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GIT for polarized curves

We investigate GIT quotients of polarized curves. More specifically, we study the GIT problem for the Hilbert and Chow schemes of curves of degree d and genus g in a projective space of dimension d-g, as the ratio v:=d/(2g-2) decreases. We prove that the first three values of v at which the GIT quotients change are given by v=4, 3.5, 2. We show that, for v>4, L. Caporaso's results hold true for both Hilbert and Chow semistability. If 3.5<v<4, the Hilbert semistable locus coincides with the Chow semistable locus and it maps to the moduli stack of weakly-pseudo-stable curves. If 2<v<3.5, the Hilbert and Chow semistable loci coincide and they map to the moduli stack of pseudo-stable curves. We also analyze in detail the critical values v=3.5 and v=4, where the Hilbert semistable locus is strictly smaller than the Chow semistable locus. As an application, we obtain three compactifications of the universal Jacobian over the moduli space of stable curves, weakly-pseudo-stable curves and pseudo-stable curves, respectively.

1. Introduction 1.1. Motivation and previous works. One of the first successful applications of Geometric Invariant Theory (GIT for short), and perhaps also one of the major motivations for its development by Mumford and his co-authors (see [MFK94]), was the construction of the moduli space M g of smooth curves of genus g ≥ 2 together with its compactification M g via stable curves, carried out by Mumford ([Mum77]) and Gieseker ([Gie82]). Indeed the moduli space of stable curves was constructed as a GIT quotient of a locally closed subset of a suitable Hilbert scheme (as in [Gie82]) or Chow scheme (as in [Mum77]) parametrizing n-canonically embedded curves, for n sufficiently large. More precisely, Mumford in [Mum77] works under the assumption that n ≥ 5 and Gieseker in [Gie82] requires the more restrictive assumption that n ≥ 10. However, it was later realized that Gieseker's approach can also be extended to the case n ≥ 5 (see [HM98,Chap. 4, Sec. C] or [Mor10, Sec. 3]).
Recently, there has been a lot of interest in extending the above GIT analysis to smaller values of n, expecially in connection with the so called Hassett-Keel program whose ultimate goal is to find the minimal model of M g via the successive constructions of modular birational models of M g (see [FS11] and [AH] for nice overviews).
The first work in this direction is due to Schubert, who described in [Sch91] the GIT quotient of the locus of 3-canonically embedded curves (of genus g ≥ 3) in the Chow scheme as the coarse moduli space M ps g of pseudo-stable curves (or p-stable curves for short). These are reduced, connected, projective curves with finite automorphism group, whose only singularities are nodes and ordinary cusps, and which have no elliptic tails. Since the GIT quotient analyzed by Schubert is geometric (i.e. there are no strictly semistable objects), it is easy to see that one gets exactly the same description working with 3-canonically embedded curves inside the Hilbert scheme (see [HH,Prop. 3.13]). Later, Hasset-Hyeon have constructed in [HH09] a modular map T : M g → M ps g which on geometric points sends a stable curve onto the p-stable curve obtained by contracting all its elliptic tails to cusps. Moreover the authors of loc. cit. identified the map T with the first contraction in the Hassett-Keel program for M g .
The case of 4-canonical curves was worked out by Hyeon-Morrison in [HM10]. The GIT quotients for both the Hilbert and Chow scheme turn out to be again isomorphic to M ps g , although the Chow quotient is not anymore geometric and a more refined analysis is required.
Finally, the case of 2-canonical curves was studied by Hassett-Hyeon in [HH], where the authors described the Hilbert GIT quotient M h g and the Chow GIT quotient M c g (they are now different), as moduli spaces of h-semistable (resp. c-semistable) curves (see loc. cit. for the precise description). Moreover, they constructed a small contraction Ψ : M For some partial results on the GIT quotient for the Hilbert scheme of 1-canonically embedded curves, we refer the reader to the work of Alper, Fedorchuck and Smyth (see [AFS]).
From the point of view of constructing new projective birational models of M g , it is of course natural to restrict the GIT analysis to the locally closed subset inside the Hilbert or Chow scheme parametrizing n-canonical embedded curves. However, the problem of describing the whole GIT quotient seems very natural and interesting too. The first result in this direction is the pioneering work of Caporaso [Cap94], where the author describes the GIT quotient of the Hilbert scheme of connected curves of genus g ≥ 3 and degree d ≥ 10(2g − 2) in P d−g . The GIT quotient obtained by Caporaso in loc. cit. is indeed a modular compactification of the universal Jacobian J d,g , which is the moduli scheme parametrizing pairs (C, L) where C is a smooth curve of genus g and L is a line bundle on C of degree d. Note that recently Li and Wang in [LW] have given a different proof of the Caporaso's result for d >> 0.
Our work is motivated by the following Problem: Describe the GIT quotient for the Hilbert and Chow scheme of curves of genus g and degree d in P d−g , as d decreases with respect to g.

1.2.
Results. In order to describe our results, we need to introduce some notations. Fix an integer g ≥ 2. For any natural number d, denote by Hilb d the Hilbert scheme of curves of degree d and arithmetic genus g in P d−g := P(V ); denote by Chow d the Chow scheme of 1-cycles of degree d in P d−g and by Ch : Hilb d → Chow d the map sending a one dimensional subscheme [X ⊂ P d−g ] ∈ Hilb d to its 1-cycle. The linear algebraic group SL d−g+1 acts naturally on Hilb d and Chow d in such a way that Ch is an equivariant map; moreover, these actions are naturally linearized (see Section 4.1 for details 1 ), so it makes sense to talk about GIT (semi-,poly-)stability of a point of Hilb d and Chow d . We aim at giving a complete characterization of the GIT (semi-,poly-)stable points [X ⊂ P d−g ] ∈ Hilb d or of its image Ch([X ⊂ P d−g ]) ∈ Chow d . Our characterization of GIT (semi-, poly-) stability will require some conditions on the singularities of X and some conditions on the multidegree of the line bundle O X (1). Let us introduce the relevant definitions.
A curve X is said to be quasi-stable if it is obtained from a stable curve Y by "blowing up" some of its nodes, i.e. by taking the partial normalization of Y at some its nodes and inserting a P 1 connecting the two branches of each node. A curve X is said to be quasi-p-stable if it is obtained from a p-stable curve Y by "blowing up" some of its nodes (as before) and "blowing up" some of its cusps, i.e. by taking the partial normalization of Y at some of its cusps and inserting a P 1 tangent to the branch point of each cusp. Given a quasi-stable or a quasi-p-stable curve X, we call the P 1 's inserted by blowing up nodes or cusps of Y the exceptional components, and we denote by X exc ⊂ X the union of all of them.
A line bundle L of degree d on a quasi-stable or quasi-p-stable curve X of genus g is said to be balanced if for each subcurve Z ⊂ X the following inequality (called the basic inequality) is satisfied where |Z ∩ Z c | denotes the length of the 0-dimensional subscheme of X obtained as the scheme-theoretic intersection of Z with the complementary subcurve Z c := X \ Z.
A balanced line bundle L on X is said to be properly balanced if the degree of L on each exceptional component of X is 1. Moreover, a properly balanced line bundle L is said to be strictly balanced (resp. stably balanced) if the basic inequality (*) is strict except possibly for the subcurves Z such that Z ∩ Z c ⊂ X exc (resp. such that Z or Z c is entirely contained in X exc ). Our first main result extends the description of GIT semistable (resp. polystable, resp. stable) points [X ⊂ P d−g ] ∈ Hilb d given by Caporaso in [Cap94] to the case d > 4(2g − 2) and also to the Chow scheme.
In each of the above cases, X ⊂ P d−g is non-degenerate and linearly normal, and O X (1) is non-special.
where Ψ ps is universally closed and surjective and Φ ps is projective, surjective and has equidimensional fibers of dimension g. (4) If char(k) = 0 or char(k) = p > 0 is bigger than the order of the automorphism group of any p-stable curve of genus g, then for any X ∈ M ps g , the fiber (Φ ps ) −1 (X) is isomorphic to Jac d (X)/Aut(X), where Jac d (X) is the Simpson's compactified Jacobian of X parametrizing S-equivalence classes of rank-1, torsion-free sheaves on X that are slope-semistable with respect to ω X . (5) If 2(2g −2) < d < 7 2 (2g −2) then J The above Theorem C follows by combining Theorems 12.3, 12.12, and Propositions 12.4, 12.8. A couple of comments on the above theorem are in order. First of all, the hypothesis on the characteristic of the base field k in part (4) is needed in order to guarantee that the automorphism groups of the geometric points of J ps d,g are linearly reductive. For more details, we refer the reader to the proof of Theorem 12.12 and the discussion following it. Secondly, the stack J ps d,g is never a Deligne-Mumford stack nor a proper stack and the map Ψ is never proper nor representable. The reason is that the automorphism group of each geometric point of J ps d,g contains the multiplicative group G m acting as scalar multiplication on the line bundle. It is then natural to take the Appendix 13 contains some positivity results for balanced line bundles on Gorenstein curves which are used throughout the paper and that we find interesting in their own.
Acknowledgements. The last two authors would like to warmly thank Lucia Caporaso for the many conversations on topics related to her PhD thesis [Cap94], which were crucial in this work.
We would like to thank Sivia Brannetti and Claudio Fontanari, who shared with us some of their ideas during the first phases of this work. We are grateful to Fabio Felici for pointing out a mistake in a previous version of Theorem 7.2.

Conventions.
1.1. k will denote an algebraically closed field (of arbitrary characteristic). All schemes are k-schemes, and all morphisms are implicitly assumed to respect the k-structure.

1.2.
A curve is a complete, reduced and separated scheme (over k) of pure dimension 1 (not necessarily connected). The genus g(X) of a curve X is g(X) := h 1 (X, O X ).

1.3.
A subcurve Z of a curve X is a closed k-scheme Z ⊆ X that is reduced and of pure dimension 1. We say that a subcurve Z ⊆ X is proper if Z = ∅, X.
Given two subcurves Z and W of X without common irreducible components, we denote by Z ∩ W the 0-dimensional subscheme of X that is obtained as the schemetheoretic intersection of Z and W and we denote by |Z ∩ W | its length.
Given a subcurve Z ⊆ X, we denote by Z c := X \ Z the complementary subcurve of Z and we set k Z = k Z c := |Z ∩ Z c |.
1.4. An elliptic tail of a curve X is a connected subcurve Z of genus one meeting the rest of the curve in one point; i.e. a connected subcurve Z ⊆ X such that g(Z) = 1 and k Z = |Z ∩ Z c | = 1.
A tacnode with a line of a curve X is a tacnode p of X at which two irreducible components D 1 and D 2 of X meet with a simple tangency and in such a way that D 1 ∼ = P 1 and k D 1 = 2 (or equivalently p is the set-theoretical intersection of D 1 and D c 1 ).

1.6.
A curve X is called Gorenstein if its dualizing sheaf ω X is a line bundle.

1.7.
A family of curves is a proper, flat morphism X → T whose geometric fibers are curves. Given a class C of curves, a family of curves of C is a family of curves X → T whose geometric fibers belong to the class C. For example: if C is the class 8 of nodal curves of genus g, a family of nodal curves of genus g is a family of curves whose geometric fibers are nodal curves of genus g.

Consider a controvariant functor
from the category SCH of schemes to the category SET of sets. We say that a scheme X represents F, or that F is represented by X, if F is isomorphic to the functor of points Hom(−, X) of X, i.e. the functor that associates to a scheme T the set of morphisms Hom(T, X).
We say that a scheme X co-represents F if there exists a natural transformation of functors Φ : F → Hom(−, X) that is universal with respect to natural transformations from F to the functor of points of schemes, i.e. for any natural transformation Ψ : Given two controvariant functors F, G : SCH → SET, we say that a natural trans- (ii) for every T ∈ SCH and x, x ′ ∈ F(T ) mapping to the same element of G(T ) there exists anétale cover {T i → T } such that the restrictions of x and x ′ to T i coincide in F(T i ).
Equivalently, a local isomorphism is a natural transformation F → G that induces an isomorphism of theétale sheaves associated to F and G. Using the well-known fact that a representable functor is a sheaf for theétale topology, it follows that if F → G is a local isomorphism then a scheme X co-represents F if and only if it co-represents G.

Singular curves
The aim of this section is to collect the definitions and basic properties of some special curves that will play a key role in the sequel. which sends X ∈ M g to the p-stable curve T (X) ∈ M ps g obtained by contracting each elliptic tail to an ordinary cusp. In particular, T is an isomorphism outside the divisor ∆ 1 ⊂ M g of curves having an elliptic tail.
If g = 2 then the functor of p-stable curves of genus g is not separated (or equivalently the stack of p-stable curves is not a Deligne-Mumford stack) and therefore it does not admit a coarse moduli space. Nevertheless, Hyeon and Lee have constructed in [HL07] a projective variety M ps 2 which co-represents the functor of p-stable curves of genus 2 (in the sense of Convention 1.8) and they have defined a map T : M 2 → M ps 2 which contracts the divisor ∆ 1 to the unique p-stable rational curve with two cusps. In this paper, however, we will often assume that g ≥ 3 whenever we will deal with p-stable curves, in order to avoid these technical issues.
2.2. wp-stable curves and p-stable reduction. A common generalization of stable and p-stable curves is provided by the wp-stable (=weakly-pseudo-stable) curves, which were first considered in [HM10, Pag. 8].
Definition 2.3. A connected curve X of genus g ≥ 2 is said to be wp-stable if (i) X has only nodes and cusps as singularities; (ii) the canonical sheaf ω X is ample.
As before, the condition that ω X is ample is equivalent to the fact that each connected subcurve Z of X of genus zero is such that k Z = |Z ∩ Z c | ≥ 3 Remark 2.4. Note that stable curves and p-stable curves are wp-stable. More precisely: (i) stable curves are exactly those wp-stable curves without cusps. (ii) p-stable curves are exactly those wp-stable curves without elliptic tails.
Given a wp-stable curve Y it is possible to obtain a p-stable curve, called its p-stable reduction and denoted by ps(Y), by contracting the elliptic tails of Y to cusps. The p-stable reduction works even for families.
Proposition 2.5. Let v : Y → S be a family of wp-stable curves of genus g ≥ 3. There exists a commutative diagram where ps(v) : ps(Y) → S is a family of p-stable curves of genus g. The family ps(v) : For every geometric point s ∈ S, the morphism ψ s : Y s → ps(Y) s contracts the elliptic tails of Y s to cusps of ps(Y) s . Moreover, the formation of ps(v) commutes with base change.
Proof. If v : Y → S is a family of stable curves, this is exactly the content of [HH09,Sec. 3]. In what follows we will show how to generalize Hassett-Hyeon's argumentation in order to work out in our case.
First of all, if S = k, then the statement follows from Proposition 3.1 in [HH09], which asserts that given a stable curve C of genus g ≥ 3, there is a replacement morphism ξ C : C → T (C), where T (C) is a pseudo-stable curve of genus g, which is an isomorphism away from the loci of elliptic tails and that replaces elliptic tails with cusps. The argumentation is local on the nodes connecting each genus-one subcurve meeting the rest of the curve in a single node. Since in a wp-stable curve all elliptic tails are connected to the rest of the curve via a single node, the same argumentation works also in our case with no further modifications.
The whole question is now how to make it work over an arbitrary base S. Hassett and Hyeon's approach is then to consider the moduli stack of stable curves M g and a faithfully flat atlas V → M g . The general case follows then by base-change from V → M g to S.
In our case, we consider the stack M wp g whose sections over a scheme S consist of families of wp-stable curves of genus g over S.
CLAIM: M wp g is a smooth, irreducible algebraic stack. 3 Indeed, M wp g is algebraic since it is an open substack of the stack of all genus g curves, which is well known to be algebraic (see e.g. [Hal] g of smooth curves is dense. Since M g is irreducible (by [DM69]), we deduce that M wp g is irreducible as well. Let now ρ π : U → M wp g be a faithfully flat atlas of M wp g and let π : Z → U be the associated (universal) family of wp-stable curves. The idea is now to consider an invertible sheaf L on Z, which will be a twisted version of the relative dualizing sheaf of π such that L is very ample away from the locus of elliptic tails, and instead has relative degree 0 over all elliptic tails. Then use L to define an S-morphism from Z to a family of p-stable curves which coincides with the previous one over all geometric fibers of π.
To be precise, denote by δ 1 ⊂ M wp g,1 the boundary divisor of elliptic tails on the universal stack M wp g,1 over M wp g . An argument similar to the proof of the above Claim shows that M wp g,1 is smooth; hence δ 1 is a Cartier divisor. Let µ π : Z → M wp g,1 be the classifying morphism corresponding to the family π : Z → U and set L := ω π (µ * π δ 1 ). The whole point is now to prove that π * (L n ) is locally free and that L n is relatively globally generated for n > 0 and that the associated morphism factors through where T (Z) is a family of p-stable curves and ξ Z coincides with the replacement morphism ξ C for all geometric fibers C of π. By browsing carefully through Hassett-Hyeon's argumentation, we easily conclude that everything holds also in our case.
2.3. Quasi-wp-stable curves and wp-stable reduction. The most general class of singular curves that we will meet throughout this work is the one given in the following: Definition 2.6.
(i) A connected curve X is said to be pre-wp-stable if the only singularities of X are nodes, cusps or tacnodes with a line. (ii) A connected curve X is said to be pre-p-stable if it is pre-wp-stable and it does not have elliptic tails. (iii) A connected curve X is said to be pre-stable if the only singularities of X are nodes.
Note that wp-stable (resp. p-stable, resp. stable) curves are pre-wp-stable (resp. pre-p-stable, resp. pre-stable) curves. Moreover, if p ∈ X is a tacnode with a line lying in D 1 ∼ = P 1 and D 2 as in 1.5, then (ω X ) |D 1 = O D 1 , hence ω X is not ample. From this, we get easily that Remark 2.7. X is wp-stable (resp. p-stable, resp. stable) if and only if X is pre-wpstable (resp. pre-p-stable, resp. pre-stable) and ω X is ample.
The pre-wp-stable curves that we will meet in this paper, even when non wp-stable, will satisfy a very strong condition on connected subcurves where the restriction of the canonical line bundle is not ample, i.e., on connected subcurves of genus zero that meet the complementary subcurve in less than three points. This justifies the following Definition 2.8. A pre-wp-stable curve X is said to be (i) quasi-wp-stable if every connected subcurve E ⊂ X such that g E = 0 and k E ≤ 2 satisfies E ∼ = P 1 and k E = 2 (and therefore it meets the complementary subcurve E c either in two distinct nodal points of X or in one tacnode of X). (ii) quasi-p-stable if it is quasi-wp-stable and pre-p-stable. (iii) quasi-stable if it is quasi-wp-stable and pre-stable.
The subcurves E such E ∼ = P 1 and k E = 2 are called exceptional and the subcurve of X given by the union of the exceptional subcurves is denoted by X exc . The complementary subcurve X c exc = X \ X exc is called the non-exceptional subcurve and is denoted by X.
Equivalently, a quasi-wp-stable curve is a pre-wp-stable X such that ω X is nef (i.e. it has non-negative degree on every subcurve of X) and such that all the connected subcurves E ⊆ X such that deg E ω X = 0 (which are called exceptional components) are irreducible. Note that the term quasi-stable curve was introduced in [Cap94, Sec.

3.3].
We summarize the different types of curves that we have so far introduced into the following Given a quasi-wp-stable curve Y , it is possible to contract all the exceptional subcurves in order to obtain a wp-stable curve, which is called the wp-stable reduction of Y and is denoted by wps(Y). This construction indeed works for families.

13
Proposition 2.9. Let S be a scheme and u : X → S a family of quasi-wp-stable curves. Then there exists a commutative diagram where wps(u) : wps(X ) → S is a family of wp-stable curves, called the wp-stable reduction of u.
For every geometric point s ∈ S, the morphism φ s : X s → wps(X ) s contracts the exceptional subcurves E of X s in such a way that The formation of wps(u) commutes with base change. Furthermore, if u is a family of quasi-p-stable (resp. quasi-stable) curves then wps(u) is a family of p-stable (resp. stable) curves.
Proof. We will follow the same ideas as in the proof of [Knu83, Prop. 2.1] and of [Mel11, Prop. 6.6]. Consider the relative dualizing sheaf ω u := ω X /S of the family u : X → S. It is a line bundle because the geometric fibers of u are Gorenstein curves by our assumption. From Corollary 13.7 in the Appendix we get that for all i ≥ 2, the restriction of ω i u to a geometric fiber X s is non-special, globally generated and, if i ≥ 3, normally generated. Then, we can apply [Knu83, Cor. 1.5] to get the following properties for ω u : , for all i ≥ 0. By (a) and (b) above, we know that S i is locally free and flat on S for i ≥ 2. Consider Since, by (d) above, the natural map is surjective, we get that there is a natural S-morphism Denote by Y the image of X via q and by φ the (surjective) S-morphism from X to Y. By (e) above, we get that So, φ : Y → S is flat since the S i 's are flat for i ≥ 2. To conclude that Y → S is a family of wp-stable curves note that the restriction of ω 3 u to the geometric fibers of u has positive degree in all irreducible components except the exceptional ones, where it has degree 0. Indeed, it is easy to see (see for example [Cat82, Rmk. 1.20]) that, on each geometric fiber X s , φ contracts an exceptional component E ⊆ X s to a node if E meets the complementary curve in two distinct nodal points and to a cusp if E meets the complementary subcurve in one tacnode. Moreover, Φ is an isomorphism outside the non exceptional locus. We conclude that Y → S is a family of wp-stable curves, so we set wps(X ) := Y and wps(u) := Y → S. Property (c) above implies that forming wps is compatible with base-change.
The last assertion is clear from the above geometric description of the contraction φ s : X s → wps(X ) s on each geometric point of u.
Remark 2.10. If u : X → S is a family of quasi-stable curves then the wp-stable reduction wps(u) : wps(X ) → S coincides with the usual stable reduction s(u) of u (see e.g. [Knu83]).
The wp-stable reduction allows us to give a more explicit description of the quasiwp-stable curves.
Corollary 2.11. A curve X is quasi-wp-stable (resp. quasi-p-stable, resp. quasistable) if and only if can be obtained from a wp-stable (resp. p-stable, resp. stable) curve Y via an iteration of the following construction: (i) Normalize Y at a node p and insert a P 1 meeting the rest of the curve in the two branches of the node. (ii) Normalize Y at a cusp and insert a P 1 tangent to the rest of the curve at the branch of the cusp.
In this case, Y = wps(X). In particular, given a wp-stable (resp. p-stable, resp. stable) curve Y there exists only a finite number of quasi-wp-stable (resp. quasi-p-stable, resp. quasi-stable) curves X such that wps(X) = Y, which we call quasi-wp-stable (resp. quasi-p-stable, resp. quasi-stable) models of Y .
Note that the above operation (ii) cannot occur for quasi-stable curves. With a slight abuse of terminology, we call the above operation (i) (resp. (ii)) the blow-up of a node (resp. of a cusp).
Proof. We will prove the Corollary only for quasi-wp-stable curves. The remaining cases are dealt with in the same way.
Let X be a quasi-wp-stable curve and set Y := wps(X). By Proposition 2.9, the wp-stablization φ : X → Y = wps(X) contracts each exceptional component E of X to a node or a cusp according to whether E ∩ E c consists of two distinct points or one point with multiplicity two. Therefore X is obtained from Y by a sequence of the two operations (i) and (ii).
Conversely, if X is obtained from a wp-stable curve Y by a sequence of operations (i) and (ii), then clearly X is quasi-wp-stable and Y = wps(X).
The last assertion is now clear.
We end this section with an extension of the p-stable reduction of Proposition 2.5 to families of quasi-wp-stable curves.
Definition 2.12. Let S be a scheme and u : X → S be a family of quasi-wp-stable curves of genus g ≥ 3. Then there exists a commutative digram v v l l l l l l l l l l l l l l l l S where the family wps(u) is the wp-stable reduction of the family u (see Proposition 2.9) and the family ps(wps(u)) is the p-stable reduction of the family wps(u) (see Proposition 2.5).
We set ps(u) := ps(wps(u)) and we call it the p-stable reduction of u.

Combinatorial results
The aim of this section is to collect all the combinatorial results that will be used in the sequel.
3.1. Balanced multidegree and the degree class group. Let us first recall some combinatorial definitions and results from [Cap94,Sec. 4]. Although the results in loc. cit. are stated for nodal curves, a close inspection of the proofs reveals that the same results are true more in general for Gorenstein curves.
So, we fix a connected Gorenstein curve X of genus g ≥ 2 and we denote by C 1 , . . . , C γ the irreducible components of X. A multidegree on X is an ordered γtuple of integers We denote by |d| = γ i=1 d C i the total degree of d. Given a subcurve Z ⊆ X, we set d Z := C i ⊆Z d C i . The term multidegree comes from the fact that every line bundle L on X has a multidegree degL given by degL := (deg C 1 L, . . . , deg Cγ L) whose total degree |degL| is the degree degL of L.
We now introduce an inequality condition on the multidegree of a line bundle which will play a key role in the sequel.
Definition 3.1. Let d be a multidegree of total degree |d| = d. We say that d is balanced if it satisfies the inequality (called basic inequality) for every subcurve Z ⊆ X.
We denote by B d X the set of all balanced multidegrees on X of total degree d.
Following [Cap94,Sec. 4.1], we now define an equivalence relation on the set of multidegrees on X. For every irreducible component C i of X, consider the multidegree More generally, for any subcurve Z ⊆ X, we set Z := C i ⊆Z C i . Denote by Λ X ⊆ Z γ the subgroup of Z γ generated by the multidegrees C i for i = 1, . . . , γ. It is easy to see that i C i = 0 and this is the only relation among the multidegrees C i . Therefore, Λ X is a free abelian group of rank γ − 1.
Definition 3.2. Two multidegrees d and d ′ are said to be equivalent, and we write For every d ∈ Z, we denote by ∆ d X the set of equivalence classes of multidegrees of total degree d = |d|. Clearly ∆ 0 X is a finite group under component-wise addition of multidegrees (called the degree class group of X) and each ∆ d X is a torsor under ∆ 0 X .
The following two facts will be used in the sequel. The first result says that every element in ∆ d X has a balanced representative. The second result investigates the relationship between balanced multidegrees that have the same class in ∆ d X .
Fact 3.3 (Caporaso). For every multidegree d on X of total degree d = |d|, there exists For a proof see [Cap94,Prop. 4.1]. Note that in loc. cit. the above result is only stated for a nodal curve X and d = 0. However, a closer inspection of the proof shows that it extends verbatim to our case. See also [MV,Prop. 2.8] for a refinement of the above result.
Moreover, the subcurves Z i can be chosen in such a way that For a proof see [Cap94, p. 620 and p. 625]. In loc. cit., the result is stated for DM-semistable curves but the proof extends verbatim to our case.

3.2.
Stably and strictly balanced multidegrees on quasi-wp-stable curves. We now specialize to the case where X is a quasi-wp-stable curve of genus g ≥ 2 (see Definition 2.8) 4 .
Given a balanced multidegree d on X, the basic inequality (3.1) gives that d E = −1, 0, 1 for every exceptional subcurve E ⊂ X. The multidegrees such that d E = 1 on each exceptional subcurve E ⊂ X will play a special role in the sequel; hence they deserve a special name.
Definition 3.5. We say that a multidegree d on X is properly balanced if d is balanced and d E = 1 for every exceptional component E of X.
We denote by B d X the set of all properly balanced multidegrees on X of total degree d.
The aim of this subsection is to investigate the behavior of properly balanced multidegrees on a quasi-wp-stable curve X, which attain the equality in the basic inequality (3.1) relative to some subcurve Z ⊆ X. Let us denote the two extremes of the basic inequality relative to Z by The definitions below will be important in what follows.
Definition 3.6. A properly balanced multidegree d on X is said to be In the case where X is a quasi-stable curve, the above Definition 3.6(i) coincides with the definition of extremal in [Cap94, Sec. 5.2], while the Definition 3.6(ii) coincides with the definition of G-stable in [Cap94, Sec. 6.2]. Here we adopt the terminology of [BFV11, Def. 2.4].
Definition 3.7. We will say that a line bundle L on X is balanced if and only if its multidegree degL is balanced. Similarly for properly balanced, strictly balanced, stably balanced.
Remark 3.8. In order to check that a multidegree d on X is balanced (resp. strictly balanced, resp. stably balanced), it is enough to check the conditions of Definitions 3.1 and 3.6 only for the subcurves Z ⊂ X such that Z and Z c are connected. This follows easily from the following facts. If Z is a subcurve of X and we denote by {Z 1 , . . . , Z c } the connected components of Z, then (i) The upper (resp. lower) inequality in (3.1) is achieved for Z if and only if the upper (resp. lower) inequality in (3.1) is achieved for every Z i . This follows from the (easily checked) additivity relations The next result explains the relationship between stably balanced and strictly balanced line bundles.
Lemma 3.9. A multidegree d on a quasi-wp-stable curve X of genus g ≥ 2 is stably balanced if and only if d is strictly balanced and X = X \ X exc is connected.
Proof. The proof of [BFV11, Lemma 2.7] extends verbatim from quasi-stable curves to quasi-wp-stable curves.
The importance of strictly balanced multidegrees is that they are unique in their equivalence class in ∆ d X , at least among the properly balanced multidegrees.
Lemma 3.10. Let d, d ′ ∈ B d X be two properly balanced multidegrees of total degree d on a quasi-wp-stable curve X of genus g ≥ 2. If d ≡ d ′ and d is strictly balanced, then Proof. According to Fact 3.4, there exist subcurves Assume, by contradiction, that d ≡ d ′ ; hence, using (3.3), we can assume that Z := Z 1 is a proper subcurve of X. From (3.4) and the fact that d is strictly balanced, we deduce that Z ∩ Z c ⊂ X exc . Therefore, there exists an exceptional subcurve E ⊆ X exc such that one of the following four possibilities occurs: Note that in Cases (II) or (IV), we have that the intersection of E with Z or Z c consists either of two distinct points or of one point of multiplicity two. Claim: Cases (III) and (IV) cannot occur. By contradiction, assume first that case (III) occurs. Consider the subcurve Z ∪ E of X. We have clearly that Therefore, using (3.4), we have that Therefore, using (3.4), it follows that which contradicts the basic inequality (3.1) for d with respect to the subcurve Z ∪ E ⊆ X. The claim is now proved.
20 Therefore, only cases (I) or (II) can occur. Note that Note also that, in any case, we must have that E ⊆ Z = Z 1 . Using (3.5), we get that E ∩ Z c i = ∅ for any i > 1, which implies that We now evaluate (3.3) at the subcurve E: using that d E = 1 because d is strictly balanced and equations (3.6) and (3.7), we conclude that In both cases, this contradicts the assumption that d ′ is properly balanced, q.e.d.
We conclude this subsection with the following Lemma, that will be used several times in what follows.
Lemma 3.11. Let X, Y and Z be quasi-wp-stable curves of genus g ≥ 2. Let σ : Z → X and σ ′ : Z → Y be two surjective maps given by blowing down some of the exceptional components of Z. Let d (resp. d ′ ) be a properly balanced multidegree on X (resp. on Y ). Denote by d the pull-back of d on Z via σ, i.e., the multidegree on Z given on a subcurve W ⊆ Z by In a similar way, we define the pull-back d ′ of d ′ on Z via σ ′ . The following is true: it is easily seen from the fact that . Therefore, in this case, the basic inequality for d with respect to W follows from the basic inequality for d with respect to σ(W ). Part (ii): start by noticing that if every exceptional component E ⊂ Z which is contracted by σ is also contracted by σ ′ then σ ′ factors through σ, so the map τ exists. Let us now prove that in order for the map τ to exist, it is also necessary that every exceptional component E ⊂ Z which is contracted by σ is also contracted by σ ′ . By contradiction, assume that τ exists and that there exists an exceptional subcurve E ⊂ Z which is contracted by σ but not by σ ′ . Then we have that where in the last equation we have used that σ(E ′ ) is an exceptional component of Y and that d ′ is properly balanced.
Since d is equivalent to d ′ by assumption, Fact 3.4 implies that we can find subcurves From (3.8) and (3.9), we get that Denote by C 1 and C 2 the irreducible components of Y that intersect E, with the convention that C 1 = C 2 if there is only one such irreducible component of Y that meets E in two distinct points or in one point with multiplicity 2. It follows directly from the definition of W (see Section 3) that for any subcurve W ⊆ Z with complementary subcurve W c we have that 1 if E ⊆ W c and exactly one among C 1 and C 2 is a subcurve of W, −1 if E ⊆ W and exactly one among C 1 and C 2 is a subcurve of W, Using this formula, together with (3.12) and (3.11), it is easy to see that C 1 must be different from C 2 and that, up to exchanging C 1 with C 2 , there exists an integer Let us now compute d Wq . From (3.11), we get that Combining this with (3.9) and (3.10), we get that Consider now the subcurve σ(W q ) of X. By (3.14), we have that and by (3.13) we have that This contradicts the fact that d is strictly balanced, q.e.d.

Preliminaries on GIT
4.1. Hilbert and Chow schemes of curves. Fix, throughout this paper, two integers d and g ≥ 2 and write d := v(2g − 2) = 2v(g − 1) for some (uniquely determined) rational number v. Set r + 1 := d − g + 1 = (2v − 1)(g − 1). Let Hilb d the Hilbert scheme parametrizing subschemes of P r = P(V ) having Hilbert polynomial P (m) := md + 1 − g, i.e., subschemes of P r of dimension 1, degree d and arithmetic genus g. An element [X ⊂ P r ] of Hilb d is thus a 1-dimensional scheme X of arithmetic genus g together with an embedding X It is well-known (see [MS,Lemma 2.1]) that for any m ≥ M := d 2 + 1 − g and any [X ⊂ P r ] ∈ Hilb d it holds that: • O X (m) has no higher cohomology; • The natural map is surjective.

23
Under these hypothesis, the m-th Hilbert point of [X ⊂ P r ] ∈ Hilb d is defined to be The ample cone of Hilb admits a finite decomposition into locally-closed cells, such that the stable and the semistable locus are constant for linearizations taken from a given cell [

In particular, there is a natural morphism of GIT-quotients
Note also that in general there is no obvious relationship between Hilbert and Chow polystability. Let ρ : G m → GL(V) be a 1ps and let x 0 , . . . , x r be coordinates of V that diagonalize the action of ρ, so that for i = 0, . . . , r we have where w i ∈ Z and w 0 ≥ . . . ≥ w r = 0. The total weight of ρ is by definition Given a monomial B = x β 0 0 . . . x βr r , we define the weight of B with respect to ρ to be For any m ≥ M as in Section 4.1 and any 1ps ρ of GL(V), we introduce the following function where the minimum runs over all the collections of P (m) monomials {B 1 , . . . , B P (m) } ∈ Sym m V ∨ which restrict to a basis of H 0 (X, O X (m)). It is easy to check that W X,ρ (m) coincide with the filtered Hilbert function of [HH,Def. 3.15]. In the sequel, we will often write W ρ (m) instead of W X,ρ (m) when there is no danger of confusion. The Hilbert-Mumford's numerical criterion for m-th Hilbert (semi)stability translates into the following (see [Gie82,p. 10] and also [HM98,Prop. 4.23]).
Indeed, the function µ([X ⊂ P r ] m , ρ) introduced above coincides with the Hilbert- The function W X,ρ (m) also allows to state the numerical criterion for Chow (semi)stability. According to [Mum77,Prop. 2.11] (see also [HH,Prop. 3.16]), the function W X,ρ (m) is an integer valued polynomial of degree 2 for m ≫ 0. We define e X,ρ (or e ρ when there is no danger of confusion) to be the normalized leading coefficient of W X,ρ (m), i.e., Remark 4.4. Observe that 2d · w(ρ) r+1 is the normalized leading coefficient of the polynomial w(ρ) r+1 mP (m) = w(ρ) r+1 m(dm + 1 − g). Therefore, combining Fact 4.3 and Fact 4.2 for m ≫ 0, one gets a proof of Fact 4.1.

Basins of attraction.
Basins of attraction represent a useful tool in the study of the orbits which are identified in a GIT quotient. We review the basic definitions, following the presentation in [HH,Sec. 4].
is Hilbert semistable (resp. Chow semistable) and is identified with [X 0 ⊂ P r ] in the GIT quotient Hilb ss d /SL r+1 (resp. Ch −1 (Chow ss d )/SL r+1 ). The following well-known properties of the basins of attraction (see e.g. [HH,) will be used in the sequel.

4.4.
Stability of smooth curves and Potential stability. Here we recall two basic results due to Mumford and Gieseker: the stability of smooth curves of high degree and the (so-called) potential stability theorem.  (i) X is pre-stable, i.e. it is reduced and has at most nodes as singularities; (ii) X ⊂ P r is non-degenerate, linearly normal (i.e., X is embedded by the complete Remark 4.9. The hypothesis that d > 4(2g − 2) in Fact 4.8 is sharp: in [HM10] it is proved that all the 4-canonical p-stable curves (which in particular can have cusps) belong to Hilb s 4(2g−2) .

Potential pseudo-stability theorem
The aim of this section is to generalize the Potential stability theorem (see Fact 4.8) for lower values of d. The main result is the following theorem, which we call potential pseudo-stability Theorem for its relations with the pseudo-stable curves (see Definition 2.1(ii)).
it is reduced and its singularities are at most nodes, cusps and tacnodes with a line. (ii) X ⊂ P r is non-degenerate, linearly normal (i.e., X is embedded by the complete Proof. The proof is an adaptation of the results in [Mum77], [Gie82], [Sch91] and [ By the inequality (5.7) of [Mum77, Pag. 96], whose proof works without any assumption on d, we get that where d = v(2g − 2). Combining the above inequalities (5.1) and (5.2) and using our assumption that v > 2, we get that is very ample, we must have that C 1 ∼ = P 1 and O C 1 (1) = O P 1 (1). However, H 1 (P 1 , O P 1 (1)) = 0 and we get a contradiction. The following result gives an answer to the above question.
Proposition 5.4. Let X be a pre-wp-stable curve of genus g ≥ 2. If there exists an ample balanced line bundle L on X of degree d ≥ g − 1 then X is quasi-wp-stable and L is properly balanced.
Proof. Let Z be a connected rational subcurve of X (equivalently Z is a chain of P 1 's) such that k Z ≤ 2. Clearly k Z ≥ 1 since X is connected and Z = X because g ≥ 2.
If k Z = 1 then deg Z (ω X ) = −1 and the basic inequality (3.1) together with the hypothesis that d ≥ g − 1 gives that This contradicts the fact that L is ample. If k Z = 2 then deg Z (ω X ) = 0 and the basic inequality (3.1) gives that Since L is ample, it has positive degree on each irreducible component of Z; therefore Z must be irreducible which implies that Z ∼ = P 1 and deg Z L = 1.
Combining the previous Proposition 5.4 with the potential stability Theorem (see Fact 4.8) and the potential pseudo-stability Theorem 5.1, we get the following  Note that, by Proposition 13.3(ii) of the Appendix, we have the following Remark, which can be seen as a partial converse to Proposition 5.4.
Remark 5.6. A balanced line bundle of degree d > 3 2 (2g − 2) on a quasi-wp-stable curve X is properly balanced if and only if it is ample. Therefore, for d > 3 2 (2g − 2), the set B d X is the set of all the multidegrees of ample balanced line bundles on X.

Elliptic tails
The aim of this section is to prove the following Theorem 6.1. If 2(2g − 2) < d < 7 2 (2g − 2) and [X ⊂ P r ] ∈ Ch −1 (Chow ss d ) ⊂ Hilb d with X connected, then X does not have elliptic tails.
Proof. Assume that 2(2g − 2) < d < 7 2 (2g − 2) and let [X ⊂ P r ] ∈ Ch −1 (Chow ss d ) ⊂ Hilb d with X connected. We known that X is quasi-wp-stable by Corollary 5.5(ii) and L := O X (1) is very ample, non special and balanced of degree d by the potential pseudo-stability Theorem 5.1.
Suppose that X has an elliptic tail, i.e. we can write X = Y ∪ F where F ⊆ X is a connected subcurve of arithmetic genus 1, Y ⊆ X is a connected subcurve of arithmetic genus g − 1 and F ∩ Y = {p} where p is a nodal point of X which is smooth in F and Y . We want to show, by contradiction, that [X ⊂ P r ] ∈ Ch −1 (Chow ss d ). Note that since Ch −1 (Chow ss d ) is open in Hilb d , we can assume that F is a generic connected curve of arithmetic genus one, and in particular that it is a smooth elliptic curve.
Let ν := degL |F . Since L (and hence L |F ) is very ample by construction, we must have ν ≥ 3. On the other hand, by applying the basic inequality (3.1) to the subcurve F ⊆ X we get which together with our assumptions on d gives that ν ≤ 3. We conclude that ν = 3. Since F is an elliptic curve, we can write for some (uniquely determined) q ∈ F . By our generic assumption on F , we can assume that q = p. Consider now the linear spans V F := F of F and V Y := Y of Y on P r = P(V ). It follows from Riemann-Roch theorem, using that L (hence L |Y and L |F ) is non special, that V F has dimension 2 and V Y has dimension d − 3 − (g − 1) = r − 2. Therefore, we can choose coordinates {x 1 , . . . , x r+1 } of V such that V F = {x 4 = . . . = x r+1 = 0}, V Y = {x 1 = x 2 = 0} and p is the point where all the x i 's vanish except x 3 . For 1 ≤ i ≤ 3, we will confound x i with the section of H 0 (F, L |F ) it determines and we will denote by ord p (x i ) the order of vanishing of x i at p. By Riemann-Roch theorem applied to the line bundles L |F (−ip) for i = 0, . . . , 3 and using (6.1) with q = p, we may choose the first three coordinates {x 1 , . . . , x 3 } of V in such a way that Consider the one parameter subgroup ρ : G m → GL(V) which, in the above coordinates, has the diagonal form ρ(t) · x i = t w i x i for i = 1, . . . , r + 1, with weights w i such that The total weight of ρ is equal to (6.4) w(ρ) = 1 + 2 + 3(r + 1 − 2) = 3r.
We want now to compute the polynomial W X,ρ (m) defined in (4.1). To that aim, consider the filtration of H 0 (X, L m ) (for m >> 0): where F r is the subspace of H 0 (X, L m ) generated by the image of the monomials B of weight w ρ (B) ≤ r via the surjection µ m : . By the definition (4.1) of W X,ρ (m), it follows that (6.5) We need to compute the dimension of F r for r < 3m. Note that if a monomial B of degree m in the above basis of V has total weight w ρ (B) < 3m then it must contain at least one factor x i with i ≤ i ≤ 2 by (6.3). Hence such a B vanishes identically on Y by our choice of the coordinates on V . Moreover, if such B contains also one factor x j with j ≥ 4, then B vanishes identically also on F , hence on the entire curve X; or in other words µ m (B) = 0. This discussion shows that F r for r < mν is mapped isomorphically via the restriction map H 0 (X, L m ) → H 0 (F, L m |F ) onto the subspace W r ⊆ H 0 (F, L m |F ) generated by the image of the monomials in {x 1 , x 2 , x 3 } of degree m and weight at most r via the multiplication map τ m : In particular (6.6) dim F r = dim W r for r < 3m.
Similarly to (4.1), define now the following function where the minimum runs over all the collections of 3m monomials {B 1 , . . . , B 3m } ∈ k[x 1 , x 2 , x 3 ] m which restrict to a basis of H 0 (F, L m |F ). As in (6.5) above, we have that (6.8) Combining (6.5), (6.6), (6.8), we get that In particular, the normalized leading coefficient e ρ of W ρ (m) is given by where e F ρ is the normalized leading coefficient of the degree 2 polynomial W F ρ (m). In order to compute the polynomial W F ρ (m), and in particular its normalized leading coefficient e F ρ , consider the embedding of F as a cubic curve in P 2 = P(H 0 (F, L |F ) ∨ ) given by the complete linear system |L |F | . Let f ∈ k[x 1 , x 2 , x 3 ] 3 be a homogenous polynomial of degree 3 defining F . The conditions (6.2) on the order at p of the coordinates {x 1 , x 2 , x 3 } translate directly into conditions on the polynomial f . More specifically, the point p has coordinates (0, 0, 1) and p ∈ F if and only if the coefficient of x 3 3 in f is equal to zero. The condition that ord p x 1 ≥ 2 says that the tangent space of F at p must have equation {x 1 = 0} which translates into the fact that the coefficient of x 2 3 x 2 in f is zero while the coefficient of x 2 3 x 1 is not zero. Finally we have that ord p (x 1 ) = 2 (i.e. p is not a flex point of F ) if and only if the coefficient of x 2 2 x 3 in f is not zero. Summing up, every polynomial f such that the coordinates {x 1 , x 2 , x 3 } satisfy (6.2) is of the form (6.11) f = a 300 x 3 1 +a 210 x 2 1 x 2 +a 201 x 2 1 x 3 +a 120 x 1 x 2 2 +a 102 x 1 x 2 3 +a 111 x 1 x 2 x 3 +a 030 x 3 2 +a 021 x 2 2 x 3 , where a 102 = 0 and a 021 = 0. Because of the choice (6.3) of the one parameter subgroup ρ, it is easy to see that the monomial x 2 2 x 3 has the maximal ρ-weight among all the monomials appearing in the above equation (6.11) of f . Moreover, the same monomial appears with non-zero coefficient in f . Therefore a collection of 3m monomials that compute the polynomial W F ρ (m), according to the formula (6.7), is represented by those monomials that are not divisible by x 2 2 x 3 , i.e.
Let us now look at the right hand side of the numerical criterion for Chow (semi)stability (see Fact 4.3). Using that v := d 2g − 2 < 7 2 by our assumptions on the degree d, we From this and (6.4), we compute (6.14) 2d From (6.13) and (6.14), we deduce that the chosen 1ps ρ satisfies e X,ρ > 2d w(ρ) r + 1 .
In other words, ρ violates the numerical criterion for Chow semistability of [X ⊂ P r ] (see Fact 4.3); hence [X ⊂ P r ] ∈ Ch −1 (Chow ss d ) which is the desired contradiction.

Stabilizer subgroups
Let [X ⊂ P r ] be a Chow semistable point of Hilb d with X connected and d > 2(2g − 2). Note that X is a quasi-wp-stable curve by Corollary 5.5(i), L := O X (1) is balanced and X is non-degenerate and linearly normal in P r by the potential pseudostability Theorem 5.1. Moreover: • If 4(2g − 2) < d then X is quasi-stable by Corollary 5.5(ii); • If 2(2g − 2) < d < 7 2 (2g − 2) then X is quasi-p-stable by Corollary 6.2. The aim of this section is to describe the stabilizer subgroup of an element [X ⊂ P r ] ∈ Hilb d as above. We denote by Stab GL r+1 ([X ⊂ P r ]) the stabilizer subgroup of [X ⊂ P r ] in GL r+1 , i.e. the subgroup of GL r+1 fixing [X ⊂ P r ]. Similarly, Stab PGL r+1 ([X ⊂ P r ]) is the stabilizer subgroup of [X ⊂ P r ] in PGL r+1 . Clearly Stab PGL r+1 ([X ⊂ P r ]) = Stab GL r+1 ([X ⊂ P r ])/G m , where G m denotes the diagonal subgroup of GL r+1 which clearly belongs to Stab GL r+1 ([X ⊂ P r ]).
It turns out that the stabilizer subgroup of [X ⊂ P r ] ∈ Hilb d is related to the automorphism group of the pair (X, O X (1)), which is defined as follows.
Given a variety X and a line bundle L on X, an automorphism of (X, L) is given by a pair (σ, ψ) such that σ ∈ Aut(X) and ψ is an isomorphism between the line bundles L and σ * (L). The group of automorphisms of (X, L) is naturally an algebraic group denoted by Aut(X, L). We get a natural forgetful homomorphism (7.1) F : Aut(X, L) → Aut(X) (σ, ψ) → σ whose kernel is the multiplicative group G m , acting as fiberwise multiplication on L, and whose image is the subgroup of Aut(X) consisting of automorphisms σ such that σ * (L) ∼ = L. The quotient Aut(X, L)/G m is denoted by Aut(X, L) and is called the reduced automorphism group of (X, L).
The relation between the stabilizer subgroup of an embedded variety X ⊂ P r and the automorphism group of the pair (X, O X (1)) is provided by the following well-known result.
Lemma 7.1. Given a projective embedded variety X ⊂ P r which is non-degenerate and linearly normal, there are isomorphisms of algebraic groups Proof. This result is certainly well-known to the experts. However, since we could not find a suitable reference, we sketch a proof for the reader's convenience.
Observe first that the natural restriction map H 0 (P r , O P r (1)) → H 0 (X, O X (1)) is an isomorphism because by assumption the embedding X ⊂ P r is non-degenerate and linearly normal. Therefore we identify the above two vector spaces and we denote them by V . Note that P r = P(V ∨ ) and that the standard coordinates on P r induce a basis of V , which we call the standard basis of V .
Let us now define a homomorphism Given (σ, ψ) ∈ Aut(X, O X (1)), where σ ∈ Aut(X) and ψ is an isomorphism between O X (1) and σ * O X (1), we define η((σ, ψ)) ∈ GL(V ∨ ) as the composition η((σ, ψ)) : where ψ −1 is the dual of the isomorphism induced by ψ −1 and σ * is the dual of the isomorphism induced by σ * . Let us denote by φ |O X (1)| (resp. φ |σ * O X (1)| ) the embedding of X in P r given by the complete linear series |O X (1)| (resp. by |σ * O X (1)|) with respect to the basis of H 0 (X, O X (1)) (resp. H 0 (X, σ * O X (1))) induced by the standard basis of V via the above isomorphisms. By construction, the following diagram commutes: Thus we get that η((σ, ψ)) belongs to Stab GL r+1 ([X ⊂ P r ]) ⊆ GL(V ∨ ) and η is welldefined. Conversely, we define a homomorphism as follows. An element g ∈ Stab GL r+1 ([X ⊂ P r ]) ⊆ GL r+1 = GL(V ∨ ) will send X isomorphically onto itself, and thus induces an automorphism σ ∈ Aut(X). Consider now the isomorphism where g −1 is the dual of g −1 and σ * is the isomorphism induced by σ. The isomorphism ψ induces an isomorphism ψ between O X (1) and σ * O X (1) making the following diagram commutative We define τ (g) := (σ, ψ) ∈ Aut(X, O X (1)). We leave to the reader the task of checking that the homomorphisms η and τ are induced by morphisms of algebraic groups and that they are one the inverse of the other.
The map η sends the subgroup G m ⊆ Aut(X, O X (1)) of scalar multiplications on O X (1) into the diagonal subgroup G m ⊂ GL r+1 and therefore it induces an isomorphism Aut(X, O X (1)) ∼ = Stab PGL r+1 ([X ⊂ P r ]). In Theorem 7.2 below, we describe the connected component Aut(X, L) 0 of Aut(X, L) containing the identity for the pairs we will be interested in. Recall, from Definition 2.8, that for a quasi-wp-stable curve X we denote by X exc ⊂ X the subcurve of X consisting of the union of the exceptional components E of X, i.e., the subcurves E ⊂ X such that E ∼ = P 1 and k E = 2. We denote by X := X c exc the complementary subcurve of X exc and by γ( X) the number of connected components of X.
Theorem 7.2. Let X be either a quasi-stable curve of genus g ≥ 2 or a quasi-p-stable curve of genus g ≥ 3 and let L be a properly balanced line bundle on X. Then the connected component Aut(X, L) 0 of Aut(X, L) containing the identity is isomorphic to G γ( X) m . Proof. Consider the wp-stable reduction X → wps(X) of X (see Proposition 2.9). Note that since wps(X) = Proj(⊕ i≥0 H 0 (X, ω i X )), an automorphism of X naturally induces an automorphism of wps(X), so by composing the homomorphism F (see (7.1)) with the homomorphism Aut(X) → Aut(wps(X)) induced by the wp-stable reduction, we get a homomorphism (7.5) G : Aut(X, L) −→ Aut(wps(X)).
Note that if X is quasi-stable of genus g ≥ 2 then wps(X) is stable of genus g ≥ 2 and that if X is quasi-p-stable of genus g ≥ 3 then wps(X) is p-stable of genus g ≥ 3.
In any case, we get that Aut(wps(X)) is a finite group, which is well-known for stable curves and proved in [Sch91, Proof of Lemma 5.3] for p-stable curves. Therefore the result follows from the claim below. CLAIM: Ker(G) = G γ( X) m . Recall from Proposition 2.9 that the wp-stable reduction X → wps(X) is the contraction of every exceptional component E ∼ = P 1 of X to a node or a cusp according to whether E ∩ E c consists of two nodes or one tacnode. We can factor the wp-stable reduction of X as where c : X → Y is obtained by contracting all the exceptional components E of X such that E ∩ E c consists of two nodes and Y → wps(X) is obtained by contracting all the exceptional components E of Y such that E ∩ E c consists of a tacnode. Now, since an automorphism of X must send exceptional components of X meeting the rest of X in two distinct points to exceptional components of the same type, we can factor the map G of (7.5) as This gives an exact sequence The same proof of [BFV11, Lemma 2.13] applied to the contraction map X → Y gives that (7.7) Ker(G 1 ) = G γ( X) m . Using (7.6) and (7.7), Claim 1 follows if we prove that (7.8) Im(G 1 ) ∩ Ker(G 2 ) = {id}.
In order to prove (7.8), we need first to describe explicitly Ker(G 2 ). Recall that, by construction, all the exceptional components E ∼ = P 1 of Y are such that E ∩ E c consists of a tacnode of Y and all of them are contracted to a cusp of wps(X) by the map Y → wps(X). Therefore Ker(G 2 ) consists of all the automorphisms γ ∈ Aut(Y ) such that γ restricts to the identity on Y \ ∪E where the union runs over all the exceptional subcurves E of Y . Consider one such exceptional component E ⊂ Y and let {p} = E ∩ E c . Since p is a tacnode of Y , there is an isomorphism (see [HH, Sec.

6.2])
i : where T p E is the tangent space of E at p and similarly for T p E c . Any γ ∈ Aut(Y ) preserves the isomorphism i. If moreover γ ∈ Ker(G 2 ) ⊆ Aut(Y ) then γ acts trivially on the irreducible component of E c containing p, hence it acts trivially also on T p E c . Therefore the restriction of γ ∈ Ker(G 2 ) to E will be an element φ ∈ Aut(E) that fixes p and induces the identity on T p E. If we fix an identification (E, p) ∼ = (P 1 , 0), the set of all such elements forms a subgroup of Aut(E) which is isomorphic to the additive subgroup G a of Aut(P 1 ) = PGL 2 given by all the transformations φ λ (for λ ∈ k) of the form (7.9) φ λ (z) = z λz + 1 .
Conversely, every such φ extends to an automorphism of Aut(Y ) that is the identity on E c and therefore lies on Ker(G 2 ). From this discussion, we deduce that (7.10) where the product runs over all the exceptional components E of Y . We can now prove (7.8). Take an element (σ, ψ) ∈ Aut(X, L) such that G 1 (σ, ψ) ∈ Ker(G 2 ). Consider an exceptional component E of Y ; let {p} = E ∩ E c and let C be the irreducible component of E c containing p. By (7.10) and the discussion preceding it, we get that G 1 (σ, ψ) |E = φ λ for some λ ∈ k (as in (7.9)) and G 1 (σ, ψ) |C = id C . By construction, the map c : X → Y is an isomorphism in a neighborhood of E ⊂ Y . Therefore, abusing notation, we identify E with its inverse image via c, similarly for p, and we call C ′ the irreducible component of X such that {p} = E ∩ C ′ . From the above properties of G 1 (σ, ψ), we deduce that σ |E = φ λ and σ |C = id C . Consider now X ∼ = E E c the partial normalization of X at p and let ν : X → X be the natural map. We have an exact sequence By looking at the gluing data defining line bundles on X, it is easy to check that the above automorphism σ ∈ Aut(X) acts as the identity on Pic( X) and that it acts on G a by sending µ into µ + λ. Since there exists an isomorphism ψ between σ * (L) and L by assumption, we must have that λ = 0, or in other words that σ |E = φ 0 = id E .
Since this is true for all the exceptional components E of Y , from (7.10) we get that G 1 (σ, ψ) = id and (7.8) is now proved.

Behaviour at the extremes of the Basic Inequality
Recall from Corollary 5.5(i) that if [X ⊂ P r ] ∈ Hilb d is Chow semistable with X connected and d > 2(2g−2), then X is quasi-wp-stable and O X (1) is properly balanced.
The aim of this section is to investigate the properties of the Chow semistable points [X ⊂ P r ] ∈ Hilb d such that O X (1) is stably balanced or strictly balanced (see Definition 3.7).
Our fist result is the following Let [X ⊂ P r ] ∈ Hilb s d ⊆ Hilb d with X connected and assume that d > 2(2g − 2). By the potential pseudo-stability Theorem 5.1 and Corollary 5.5(i), we get that X is a quasi-wp-stable curve and L := O X (1) is properly balanced and non-special.
By contradiction, suppose that O X (1) is not stably balanced. Then, by Definition 3.6 and Remark 3.8, we can find a connected subcurve Y with connected complementary subcurve Y c such that In order to produce the desired contradiction, we will use the numerical criterion for Hilbert stability (see Fact 4.2). Let V := H 0 (P r , O P r (1)) = H 0 (X, O X (1)) and consider the vector subspace U := Ker H 0 (P r , O P r (1)) → H 0 (Y, L |Y ) ⊆ V.
Set N + 1 := dim U . Choose a basis {x 0 , . . . , x N , . . . , x r } of V relative to the filtration U ⊆ V , i.e., x i ∈ U if and only if 0 ≤ i ≤ N . Define a 1ps ρ of GL r+1 by We will estimate the two polynomials appearing in Fact 4.2 for the 1ps ρ.
First of all, the total weight w(ρ) of ρ satisfies w(ρ) In order to compute the polynomial W ρ (m) for m ≫ 0, consider the filtration of H 0 (P r , O P r (m)): where U m−i V i is the subspace of H 0 (P r , O P r (m)) generated by the monomials containing at least (m − i)-terms among the variables {x 0 , . . . , x N }. Note that for a monomial B of degree m, it holds that Via the surjective restriction map µ m : H 0 (P r , O P r (m)) ։ H 0 (X, L m ), the above filtration on H 0 (P r , O P r (m)) induces a filtration It remains to estimate dim F i for 0 ≤ i ≤ m − 1. To that aim, consider the partial normalization τ :X → X of X at the nodes laying on Y ∩ Y c . Observe thatX is the disjoint union of Y and Y c . We denote by Consider now the injective pull-back morphism which clearly coincides with the restriction maps to Y and Y c . Note that if B is a monomial belonging to U m−i V i ⊆ H 0 (P r , O P r (m)) for some i ≤ m − 1, then B contains at least m − i ≥ 1 variables among the x j 's such that x j ∈ U ; hence the order of vanishing of B along the subcurve Y is at least equal to m − i . This implies that any s ∈ F i ⊆ H 0 (X, L m ) with i ≤ m − 1 vanishes identically on Y and vanishes on the points of D Y c with order at least (m − i). We deduce that Let us prove the claim. Clearly if the claim is true for i = 0 then it is true for every i > 0; so we can assume that i = 0. According to Fact 13.4(i) of the Appendix, it is enough to prove that for any connected subcurve Z ⊆ Y c , we have that where k Z is, as usual, the length of the schematic intersection of Z with the complementary subcurve X \ Z in X. In order to prove (8.7), we consider different cases. If g Z ≥ 1 then using the basic inequality (3.1) for L relative to the subcurve Z and the assumption d > 2(2g − 2), we compute which shows that (8.7) holds in this case. If g Z = 0 and Z = Y c then, using that deg Y c L = M Y c and k Y c ≥ 3 by (8.1), we get which shows that (8.7) holds also in this case. It remains to consider the case g Z = 0 and Z Y c . If k Z ≤ 2 then, since X is quasi-wp-stable and Z is connected, we must have that Z is an exceptional subcurve of X, i.e., Z ∼ = P 1 and k Z = 2. By Proposition 5.4 it follows that deg Z L = 1. Since |D Z | ≤ 1 by (8.8), we deduce that (8.7) is satisfied also in this case. Finally, assume that k Z ≥ 3. Consider the subcurve W := Z c ∩ Y c ⊂ Y c . It is easy to check that Using the basic inequality of L with respect to W together with (8.1), (8.9) and k Z ≥ 3, we get The claim is now proved. Using the claim above, we get from (8.5) that (8.10) Combining (8.10) and (8.4), we get that where in the last equality we have used Using that deg Y L = m Y by (8.1), we easily check that By combining (8.2), (8.11), (8.12), (8.13), we get for m ≫ 0: which contradicts the numerical criterion for Hilbert stability (see Fact 4.2), q.e.d.
We start by introducing an order relation on the set of pairs (X, L) where X is a quasi-wp-stable curve and L is a properly balanced line bundle on X of degree d. (ii) We say that (X ′ , L ′ ) is an isotrivial specialization of (X, L), and we write (X, L) There is a close relationship between the existence of isotrivial specializations and strictly balanced line bundles, as explained in the following Proof. Part (i). Clearly, it is enough to consider the case where (X, L) el (X ′ , L ′ ) is an elementary isotrivial specialization as in Definition 8.3(i). For Z ⊆ X ′ as in Definition 8.3(i), decompose Z c as the union of all the exceptional components {E i } i=1,··· ,k Z of X ′ that meet Z and a subcurve W . By applying Remark 3.8(i) to the subcurve E 1 ∪· · ·∪E k Z , where the basic inequality achieves its maximal value, it is easy to see that deg W L ′ = m W . Let now W be the subcurve of X given by the union of the irreducible components of X that specialize to an irreducible component of W ⊂ X ′ . Since (X, L) is obtained from (X ′ , L ′ ) by smoothing some nodes which belong to Z ∩ ∪ i E i and therefore are not in W , we clearly have that W ∼ = W , k W = k W and L W ∼ = L ′ W . Hence deg W L ′ = m W and, since W ∩ W c ⊂ X exc , we conclude that L is not strictly balanced.
Part (ii). If L is not strictly balanced, we can find a subcurve Y ⊂ X such that Let us denote by {n 1 , . . . , n r } the points belonging to Y ∩ Y c \ X exc . Let X ′ be the blow-up of X at {n 1 , . . . , n r } and let E Y := E 1 ∪ · · · ∪ E r be the new exceptional components of X ′ . Given a subcurve Z ⊆ X denote by Z ′ the strict transform of Z via the blow-up morphism and define k Y 1, for i = 1, . . . , r and, given an irreducible component C of X, where X ′ is the blow-up of X at {n 1 , . . . , n r } and degL ′ = d.
Let us check that L ′ is properly balanced. It is clear that the degree of L ′ on all the exceptional components of X ′ is equal to one. Let W ⊆ X ′ and let us check that L ′ satisfies the basic inequality (3.1). Start by assuming that W = Z ′ for some Z ⊆ Y . Then we have that Analogously we can show that deg W L ′ ≤ M W , so we conclude that L ′ is properly balanced. Now, if L ′ is strictly balanced we are done. If not, we repeat the same procedure and after a finite number of steps we will find the desired pair (X ′′ , L ′′ ) with L ′′ strictly balanced.
We can now give a partial answer to Question 8.2.
We can clearly assume that (X, O X (1)) el (X ′ , O X ′ (1)). Using the notations of Definition 8.3(i), this means that there exists a connected subcurve Z ⊂ X ′ with Z c connected and Z ∩ Z c ⊂ X ′ exc and deg Z L ′ = m Z such that (X, O X (1)) is obtained from (X ′ , O X ′ (1)) by smoothing some of the nodes of Z ∩ Z c . Moreover, we can decompose the connected complementary subcurve Z c as where the E i 's are the exceptional subcurves of X ′ that meet the subcurve Z and W := Z c \ ∪ i E i is clearly connected as well. Since deg E i L ′ = 1, it follows from Remark 3.8 applied to the subcurve The 1ps ρ of GL r+1 we are looking for is similar to the 1ps considered in the proof of Theorem 8.1. More precisely, consider the restriction map The map res is injective since the complementary subcurve of Z ∪ W is made of the exceptional components E i ∼ = P 1 , each of which meets both Z and W in one point. Moreover, since O X ′ (1) is non-special by assumption, which implies that also O Z (1) and O W (1) are non-special, we have that where we have used that m Z + m W = d − k Z and g = g W + g Z + k Z − 1. This implies that res is an isomorphism. Define now the 1ps ρ : G m → GL r+1 in such a way that Let us check that the above 1ps ρ satisfies all the desired properties.
CLAIM 2: ρ stabilizes [X ′ ⊂ P r ] ∈ Hilb d . Using Lemma 7.1, it is enough to check that Since the non exceptional subcurve X ′ ⊂ X ′ is contained in Z W , it follows from the proof of Theorem 7.2 that Aut(X ′ , O X ′ (1)) contains a subgroup H isomorphic to G 2 m and such that (λ, µ) ∈ H ∼ = G 2 m acts via multiplication by λ on H 0 (W, O W (1)) and by µ on H 0 (Z, O Z (1)). By construction, it follows that Imρ ⊆ H and we are done.
CLAIM 3: Recall that, by assumption, (X, O X (1)) is obtained from (X ′ , O X ′ (1)) by smoothing some of the nodes of Z ∩ Z c = ∪ i (Z ∩ E i ). Denote by n i the node given by the intersection of Z with E i and by Def (X ′ ,n i ) the functor of infinitesimal deformations of the complete local ring O X ′ ,n i (see [Ser06,Sec. 2.4]). According to [Ser06, Cor. 3.1.2,  Exa. 3.1.4(a) Consider now the local Hilbert functor H P r X ′ parametrizing infinitesimal deformations of X ′ in P r (see [Ser06, Sec. 3.2.1]). Clearly, H P r X ′ is pro-represented by the complete local ring of Hilb d at [X ⊂ P r ]. Since X ′ is a curve with locally complete intersection singularities and O X ′ (1) is non-special, from [Kol96, I.6.10] we get that the natural morphism of functors is formally smooth, where Def X ′ is the functor of infinitesimal deformations of X ′ . It follows easily from [Ser06, Thm. 2.4.1], that also the natural morphism of functors is formally smooth. Moreover, since ρ stabilizes [X ′ ⊂ P r ] by Claim 2, the above morphisms (8.16) and (8.17) are equivariant under the natural action of ρ on each functor. Therefore, in order to prove that [X ⊂ P r ] ∈ A ρ ([X ′ ⊂ P r ]), it is enough to prove that ρ acts on each k[[a i ]] with positive weight (compare also with the proof of [HM10, Lemma 4] and of [HH,Cor. 7.9]). Fix a node n i = E i ∩ Z for some 1 ≤ i ≤ k Z . We can choose coordinates {x 1 , . . . , x r+1 } of V = H 0 (P r , O P r (1)) = H 0 (X ′ , O X ′ (1)) in such a way that x i is the unique coordinate which does not vanish at n i , the exceptional component E i is given by the linear span x i , x i+1 and the tangent T Z,n i of Z at n i is given by the linear span x i−1 , x i . Then the completion of the local ring O X ′ ,n i is equal to On the other hand, the point q i defined by x k = 0 for every k = i + 1 is clearly the node given by the intersection of E i with W . Since ρ(t) |H 0 (W,O W (1)) = t · Id by construction, we have that ρ(t) · x i+1 = tx i+1 ; hence ρ(t) · v i = tv i . Since the equation of the universal family over k[[a i ]] is given by u i v i − a i = 0 and ρ acts on this universal family, we deduce that ρ(t) · a i = ta i , which concludes our proof.
From the above theorem, we deduce now the following Corollary 8.6. Let [X ⊂ P r ] ∈ Hilb d with X connected and d > 2(2g − 2). If [X ⊂ P r ] is Chow polystable or Hilbert polystable then O X (1) is strictly balanced.
Proof. Let us prove the statement for the Chow polystability; the Hilbert polystability being analogous.
Let [X ⊂ P r ] ∈ Hilb d for d > 2(2g − 2) with X connected and assume that [X ⊂ P r ] is Chow-polystable. Recall that X is quasi-wp-stable by Corollary 5.5(i) and O X (1) is properly balanced by Theorem 5.1 and Proposition 5.4. By Lemma 8.4, we can find a pair (X ′ , L ′ ) consisting of a quasi-wp-stable curve X ′ and a strictly balanced line bundle L ′ on X ′ such that (X, O X (1)) (X ′ , L ′ ). Note that L ′ is ample by Remark 5.6; moreover X ′ does not have elliptic tails if d < 5/2(2g − 2) because X satisfies the same property by Theorem 6.1 and in an isotrivial specialization no new elliptic tails can be created (see Definition 8.3). Therefore, we can apply Theorem 13.5 which allows to conclude that L ′ is non-special and very ample; we get a point 9. The map towards the moduli space of p-stable curves Since H d ⊂ Ch −1 (Chow ss d ) is clearly an SL(V)-invariant subscheme, GIT tells us that there exists a projective scheme which is a good categorical quotient of H d by SL(V) (see e.g. [Dol03, Sec. 6.1]).
Theorem 9.1. Assume that d > 2(2g − 2) and g ≥ 3. Then: (i) There exists a surjective natural map Φ ps : Q d,g → M ps g . (ii) If d > 4(2g − 2) then the above map Φ ps factors as where T is the map of Fact 2.2(ii). 46 (iii) We have that g is the open subset of M g parametrizing curves without non-trivial automorphisms and J 0 d,g is the degree d universal Jacobian over M 0 g . In particular, g . If d > 4(2g − 2) then the same conclusions hold for the morphism Φ s . (iv) H d is non-singular of pure dimension r(r + 2) + 4g − 3.
(v) Q d,g is reduced and normal of dimension 4g − 3. Moreover, if char(k) = 0, then Q d,g has rational singularities, hence it is Cohen-Macauly.
Indeed, the proof below will show that the morphism Φ s exists also for g = 2 (and d > 4(2g − 2)).
Proof. The proof is an adaptation of the ideas from [Cap94, Sec. 2].
Part (i): consider the restriction to H d of the universal family over Hilb d and denote it by The morphism u d is flat, proper and its geometric fibers are quasi-wp-stable curves by Corollary 6.2(ii). Consider the p-stable reduction of u d (see Definition 2.12): The morphism ps(u d ) is flat, proper and its geometric fibers are p-stable curves of genus g. Therefore, by the modular properties of M ps g , the family ps(u d ) induces a modular map φ ps : H d → M ps g . Since the group SL(V) = SL r+1 acts on the family C d by only changing the embedding of the fibers of u d into P r , the map φ ps is SL r+1 -invariant and therefore it factors via a map Φ ps : Q d,g → M ps g . Let us show that Φ ps is surjective. Let C be any connected smooth curve over k of genus g ≥ 2 and L be any line bundle on C of degree d > 2(2g − 2). Note that d = degL ≥ 2g + 1 since g ≥ 2. Hence L is very ample and non-special and therefore it embeds C in P r = P d−g . By Fact 4.7, the corresponding point [C |L| ֒→ P r ] ∈ Hilb d belongs to H d and clearly it is mapped to C ∈ M g ⊂ M ps g by Φ ps . We conclude that the image of Φ ps contains the open dense subset M g ⊂ M ps g . Moreover, Φ ps is projective since Q d,g is projective. Therefore, being projective and dominant, Φ ps has to be surjective. This finishes the proof of part (i).
Consider now Part (ii). If d > 4(2g − 2), then the potential stability Theorem (see Fact 4.8) says that the geometric fibers of the morphism u d are quasi-stable curves.
From Definition 2.12 and Proposition 2.9, it follows that the p-stable reduction ps(u d ) of u d factors through the wp-stable reduction wps(u d ) of u d and that the latter one is a family of stable curves. This implies that the map Φ ps : Q d,g → M ps g factors via a map Φ s : Q d,g → M g followed by the contraction map T : M g → M ps g . Part (iii): the proof of [Cap94, Thm. 2.1(2)] extends verbatim to our case. Part (iv): the fact that H g is non-singular of pure dimension r(r + 2) + 4g − 3 is proved exactly as in [Cap94, Lemma 2.2], whose proof uses only the fact that if X ∈ H d then X is reduced, a local complete intersection and embedded by a non-special linear system; these conditions are satisfied by the potential pseudo-stability Theorem 5.1. See also [HH,Cor. 6.3] for another proof.
Part (v): Q d,g is reduced and normal since H d is (see e.g. [Dol03,Prop. 3.1]). The dimension of Q d,g is 4g − 3 since H d has dimension r(r + 2) + 4g − 3, SL r+1 has dimension r(r + 2) and the action of SL r+1 has generically finite stabilizers. If char(k) = 0 then Q d,g has rational singularities by [Bou87], using that H d is smooth. This implies that Q d,g is Cohen-Macauly since, in characteristic zero, a variety having rational singularities is Cohen-Macauly (see [KoM98,Lemma 5.12]). Alternatively, the fact that Q d,g is Cohen-Macauly follows from [HR74], using the fact that H d is smooth.
Further properties of Q d,g and of the morphisms Φ ps and Φ s will be proved later on in Proposition 10.9.

A stratification of the semistable locus
Inspired by [Cap94, Sec. 5], we introduce in this section an SL r+1 -invariant stratification of H d (see (9.1)) and we establish some properties of it.
Recall that every X ∈ H d is quasi-wp-stable and that O X (1) is properly balanced by Corollary 5.5(i). Recall also that B d X denotes the set of multidegrees of properly balanced line bundles on X of total degree d (see Definition 3.5).
Following [Cap94,Sec. 5.1], consider, for any quasi-wp-stable curve X of genus g and any d ∈ B d X , the (locally closed) stratum Each stratum M d X is SL r+1 -invariant since SL r+1 acts on H d by changing the embedding of X inside P r and thus it preserves X and the multidegree d. Note that M d X may be empty for certain pairs (X, d) as above.
10.1. Specializations of strata. The aim of this subsection is to describe all pairs Generalizing the refinement relation of [Cap94,Sec. 5.2], we define an order relation on the sets of pairs (X, d) where X is a quasi-wp-stable curve of genus g and d ∈ B d X .
Definition 10.1. Let X ′ and X ′′ be two quasi-wp-stable curves of genus g.

48
(i) We say that X ′′ X ′ if they have the same wp-stable reduction X = wps(X ′ ) = wps(X ′′ ) and there exists a surjective morphism σ : X ′′ → X ′ commuting with their wp-stable reduction morphisms (ii) Assume moreover that d ′ ∈ B d X ′ and that d ′′ ∈ B d X ′′ . We say that (X ′′ , d ′′ ) (X ′ , d ′ ) if X ′′ X ′ and there exists a surjective morphism σ : X ′′ → X ′ as before such that for every subcurve The order relation can be described in terms of elementary operations as follows.
Lemma 10.2. With the same notations as in the above Definition 10.1, we have that if and only if X ′′ is obtained from X ′ via a sequence of blow-ups of nodes and cusps of X ′ and the the multidegree d ′′ is obtained from d ′ at each step according to the rules depicted in Figures 1 and 2. Figure 1. Blow-up of a node: external and internal cases. Proof. Using the explicit description of the wp-stable reduction of Proposition 2.9, it is easy to see that X ′′ X ′ if and only if X ′′ is obtained from X ′ via a sequence of blow-ups of nodes and cusps. According to Definition 10.1(ii), it is now clear that at each blow-up d ′′ must be obtained from d ′ according to the rules depicted in Figures 1  and 2.
From the above description it is easy to see that there is a relation between the isotrivial specialization introduced in Definition 8.3 and the order relation . More precisely, we have the following Remark 10.3. Let (X ′ , L ′ ) and (X ′′ , L ′′ ) be two pairs consisting of a quasi-wp-stable curve of genus g and a properly balanced line bundle of degree d. If (X ′ , L ′ ) (X ′′ , L ′′ ) then (X ′′ , degL ′′ ) (X ′ , degL ′ ).
The following elementary property of the order relation will be used in what follows.
. Proof. By Lemma 10.2 above it is enough to assume that X ′′ is obtained from X ′ by blowing-up a node (which can be internal or external) or a cusp. Start by assuming that X ′′ is obtained from X ′ by blowing up an external node N , as in the picture on the left of Figure 1.
Denote by {C ′ 1 , . . . , C ′ γ } the irreducible components of X ′ , by {C ′′ 1 , . . . , C ′′ γ } their proper transforms in X ′′ and by E the exceptional component that is contracted to the node N by the map σ : X ′′ → X ′ . Assume that C ′ 1 and C ′ 2 are the two irreducible components of X ′ that contain the node N . Define a multidegree d ′ on X ′ in the following way: It is clear that |d ′ | = d, so we must check that d ′ satisfies the basic inequality (3.1). Given a subcurve Z ′ of X ′ , we denote by Z ′′ the subcurve of X ′′ that is the proper transform of Z ′ under the blow-up map X ′′ → X ′ . Define W Z ′ to be the subcurve of Hence the basic inequality (3.1) for d ′ relative to the subcurve Z ′ is the same as the basic inequality for d ′′ relative to the subcurve W Z ′′ . We conclude that if d ′′ ∈ B d X ′′ then d ′ ∈ B d X ′ . The cases where X ′′ is obtained from X ′ by blowing up an internal node or a cusp are similar (and easier) and are therefore left to the reader.
We will now prove that the above order relation determines the inclusion relations among the closures of the strata M Proposition 10.5. Assume that d > 2(2g−2) and moreover that g ≥ 3 if d < 4(2g−2). Let X ′ and X ′′ be two quasi-wp-stable curves of genus g and let Proof. ⇐= From Lemma 10.2 above, it is enough to assume that X ′′ is obtained from X ′ by blowing up a node or a cusp. Assume that X ′′ is obtained from X ′ by blowing up a node, say N . Let B be a smooth curve and consider the trivial family X ′ × B over B. By blowing up the surface X × B on the node N belonging to the fiber over a point b 0 ∈ B, we get a family u : X → B whose geometric fiber X b over a point b ∈ B is such that X b ∼ = X ′ for all b = b 0 and X b 0 ∼ = X ′′ as in the figure below (where we have depicted an external node, but the case of an internal node is completely similar).
Consider the relative Picard scheme π : Pic X /B → B of the family u : X → B, which exists by a well-known result of Mumford (see [BLR90,Sec. 8 We can view L ′′ as a geometric point of (Pic X /B ) b 0 ∼ = Pic(X ′′ ). Since the morphism π : Pic X /B → B is smooth, up to shrinking B (i.e., replacing it with anétale open neighborhood of b 0 ), we can find a section σ of π such that σ(b 0 ) = L ′′ . Moreover, by definition of the order relation (see Figure 1), it is clear that we can choose the section σ in such a way that σ(b) is a line bundle of multidegree d ′ on X b ∼ = X ′ for every b = b 0 .
Up to shrinking B again, we can assume that the section σ corresponds to a line bundle L over X such that L |X b 0 ∼ = L ′′ and L |X b has multidegree d ′ for b = b 0 . Since L ′′ is very ample and non-special and these conditions are open, up to shrinking B once more, we can assume that L is relatively very ample and we can fix an isomorphism Φ : x x p p p p p p p p p p p p B whose restriction over b 0 ∈ B is the embedding X ′′ ⊂ P r . The family u : X → B together with the embedding i defines a morphism f : In the case when X ′′ is obtained from X ′ by blowing up a cusp we proceed in the same way as in the previous case: we consider a family u : X → B such that and we apply the same argument as before.
=⇒ Consider the map (see Theorem 9.1): , which implies that ps(X ′′ ) = X = ps(X ′ ). By assumption, we can find a smooth curve B and a morphism f : By pulling back the universal family above H d along the morphism f we get a family The ps-stable reduction ps(u) of u is an isotrivial family of p-stable curves by what observed before. This implies that X is obtained from an isotrivial family X with fibers isomorphic to X ′ by blowing up some nodes and cusps of the central fiber X b 0 = X ′ . We get a surjective morphism σ : X ′′ = X b 0 → X b 0 = X ′ which clearly commutes with the p-stable reduction morphisms; in other words X ′′ X ′ . Consider now the line bundles Let Y ′ ⊆ X ′ be a subcurve of X ′ . Consider the subcurve Y ′′ ⊆ X ′′ = X b 0 given by the union of all the irreducible components C i of X ′′ for which there exists a section s of u : Proof. We first make the following two reductions.
Reduction 1: We can assume that if d < 5 2 (2g − 2) then X does not contain elliptic tails; in this case every L ∈ Pic d (X) is non-special and very ample.
Indeed, according to Theorem 13.5(i), L ∈ Pic d (X) is non-special since X is quasiwp-stable, hence G-semistable, and degL = d > 2(2g − 2) > 2g − 2 (recall that g ≥ 2). Now, if d < 5 2 (2g−2) and X contains some elliptic tail F , then from the basic inequality it follows easily that d F = 2. But no line bundle of degree 2 on a curve of genus 1 is very ample, hence no line bundle of multidegree d on X can be very ample. Otherwise, since any L ∈ Pic d (X) is ample by Remark 5.6, it follows from Theorem 13.5(iii) that L is very ample, q.e.d.
Reduction 2: We can assume that d is strictly balanced. Indeed, suppose the proposition is true for all strictly balanced line bundles on quasiwp-stable curves and let us show that it is true for our multidegree d on X, assuming that d is not strictly balanced. Let L ∈ Pic d (X). Since d is not strictly balanced, by Lemma 8.4(ii) there exists an isotrivial specialization (X, L) (X ′ , L ′ ) such that d ′ := degL ′ is a strictly balanced multidegree on X ′ . Moreover, from the proof of the cited Lemma, it follows easily that the curve X ′ and the multidegree d ′ depend only on X and d and not on L ∈ Pic d (X). Note that, since X ′ is obtained from X by blowing up some nodes of X, then X has some elliptic tails if and only if X ′ has some elliptic tails. Therefore, according to Reduction 1, L and L ′ are non-special and very ample. Up to the choice of a basis of |L| and of |L ′ |, we get two points of Hilb d , [X ′ |L ′ | ֒→ P r ] ∈ Ch −1 (Chow ss d ). Therefore, Theorem 8.5(ii) gives that L ∈ Im(p) if and only if L ′ ∈ Im(p ′ ). In other words, we have defined a set-theoretic map such that Υ −1 (Im(p ′ )) = Im(p). The proposition for d ′ is equivalent to the fact that either Im(p ′ ) = ∅ or Im(p ′ ) = Pic d ′ (X ′ ). Using the above map Υ, it is easy to see that the above properties hold also for d, hence we can assume that d is strictly balanced, q.e.d.
We now prove the proposition for a pair (X, d) satisfying the properties of Reduction 1 and Reduction 2. Assume that M d X = ∅, for otherwise there is nothing to prove. Let us first prove the following CLAIM: The image of the morphism p : M d X → Pic d (X) is open and dense. Consider a Poincaré line bundle P on X × Pic d (X), i.e., a line bundle P such that P |X×{L} ∼ = L for every L ∈ Pic d (X) (see [Kle05,Ex. 4.3]). By Reduction 1, it follows that P is relatively very ample with respect to the projection π 2 : X × Pic d (X) → Pic d (X) and that (π 2 ) * (P) is locally free of rank equal to r + 1 = d − g . We can therefore find a Zariski open cover {U i } i∈I of Pic d (X) such that (π 2 ) * (P) |U i ∼ = O ⊕(r+1) U i and the line bundle P induces an embedding y y r r r r r r r r r r r U i The above embedding corresponds to a map f i : U i → Hilb d and it is clear that Since  We denote by L the relatively ample line bundle on X := X × B → B which gives the embedding into P r B . We can now apply a fundamental result in GIT, called semistable replacement property (see e.g. [HH,Thm. 4.5]), which implies that, up to replacing B with a finite cover ramified over b 0 , we can find two maps g : B → H d and h : B \ {b 0 } → SL r+1 such that This implies that the multidegrees d ′ and d on Y are equivalent in the sense of Definition 3.2. Since d is strictly balanced by Reduction 1, we can now apply Lemma 3.11 (with Z = Y and σ ′ = id) in order to conclude that X = Y or, equivalently, X = Y. Since we have already observed that (Y, d ′ ) (X, d), we must have that d = d ′ . Combining this with (10.5), we get that L :  Fact A: Let f : X → Y be a surjective morphism between two varieties. Assume that Y is irreducible. If f has equi-dimensional fibers of the same dimension and the generic fiber is irreducible, then X is irreducible.
We apply the above Fact A to the morphism Φ s and we use the fact that M g is an irreducible variety (see Fact 2.2(i)), that Φ s has equidimensional fibers of dimension g by what we have just proved and that the generic fiber of Φ s is irreducible by Theorem 9.1(iii). The irreducibility of Q d,g follows.
In order to prove the flatness of Φ s over the smooth locus (M g ) sm of M g , we will use the following well-known flatness's criterion.
Fact B (see [Mat89, Cor. of Thm 23.1, p. 179]): Let f : X → Y be a dominant morphism between irreducible varieties. If X is Cohen-Macauly, Y is smooth and f has equi-dimensional fibers of the same dimension, then f is flat.
We apply the above Fact B to the restriction morphism Φ s : (Φ s ) −1 ((M g ) sm ) → (M g ) sm and we use that Φ s has equidimensional fibers of dimension g as we proved above and that Q d,g is Cohen-Macauly if char(k) = 0 by Theorem 9.1(v).
Assume now that 2(2g − 2) < d < 7 2 (2g − 2). The proof is entirely analogous to the previous proof noticing that, by Corollary 6.2, the fiber of the morphism where the union is over the possible quasi-p-stable curves X ′ whose p-stable reduction ps(X ′ ) = wps(X ′ ) is equal to X and d ′ ∈ B d X ′ . We leave the details to the reader.

Semistable, polystable and stable points
The aim of this section is to describe the points of Hilb d that are Hilbert or Chow semistable, polystable and stable. Let us begin with the semistable points.
Theorem 11.1. Consider a point [X ⊂ P r ] ∈ Hilb d and assume that X is connected.
(1) If d > 4(2g − 2) then the following conditions are equivalent: (iii) X is quasi-stable, non-degenerate and linearly normal in P r and O X (1) is properly balanced and non-special; (iv) X is quasi-stable and O X (1) is properly balanced; (v) X is quasi-stable and O X (1) is balanced.
(2) If 2(2g − 2) < d < 7 2 (2g − 2) and g ≥ 3 then the following conditions are equivalent: (iii) X is quasi-p-stable, non-degenerate and linearly normal in P r and O X (1) is properly balanced and non-special; (iv) X is quasi-p-stable and O X (1) is properly balanced; (v) X is quasi-p-stable and O X (1) is balanced.
(1iv) ⇒ (1i) First of all, we make the following Reduction: We can assume that O X (1) is strictly balanced. Indeed, by Lemma 8.4(ii), there exists an isotrivial specialization (X, O X (1)) (X ′ , L ′ ) such that X ′ is quasi-stable and L ′ is a strictly balanced line bundle on X ′ of total degree d. According to Theorem 13.5 and using that d > 4(2g − 2), we conclude that L ′ is very ample and non-special. Therefore, by choosing a basis of H 0 (X ′ , L ′ ), we get a point [X ′ |L ′ | ֒→ P r ] ∈ Hilb d . According to Theorem 8.5, [X ⊂ P r ] ∈ Hilb ss d if and only if [X ′ ⊂ P r ] ∈ Hilb ss d . Therefore, up to replacing X with X ′ , we can assume that O X (1) is strictly balanced, q.e.d. Now, since X is quasi-stable, we can find a smooth curve B f ֒→ Hilb d and a point b 0 ∈ B such that, if we denote by P r × B i ←֓ X π → B the pull-back via f of the universal family over Hilb d and we set L := i * (O P r (1)⊠O B ), then [X i ֒→ P r ×B] b 0 = [X ⊂ P r ] and X |π −1 (b) is a connected smooth curve for every b ∈ B \{b 0 }. Note that, by construction, π is a family of quasi-stable curves of genus g. As in the proof of Proposition 10.6, we can now apply the semistable replacement property, which implies that, up to replacing B with a finite cover ramified over b 0 , we can find two maps g : B → Hilb d and h : B \ {b 0 } → SL r+1 such that (11.2) g(b 0 ) is Hilbert polystable.
We denote by P r × B Note that this fact together with (11.2) and the potential stability Theorem (Fact 4.8) implies that π ′ is also a family of quasi-stable curves of genus g.
Consider now the stable reductions s(π) : s(X ) → B of π : X → B and s(π ′ ) : s(Y) → B of π ′ : Y → B (see Remark 2.10). From (11.3), it follows that s(π) and s(π ′ ) are two families of stable curves which are isomorphic away from the fibers over b 0 . Since the moduli space M g of stable curves is separated, we conclude that (11.4) s(X ) Therefore π and π ′ are two families of quasi-stable curves with the same stable reduction (from now on, we identify s(X ) s(π) −→ B and s(Y) s(π ′ ) −→ B via the above isomorphism). If we blow-up all the nodes of the fiber over b 0 of the stable reduction s(π) = s(π ′ ), we get a new family of quasi-stable curves π : Z → B with the same stable reduction as that of π and of π ′ , which moreover dominates π and π ′ , i.e., such that there exists a commutative diagram (11.5) Z π Σ ′ ? ? ?
where the maps Σ and Σ ′ induce an isomorphism of the corresponding stable reductions. Equivalently, the maps Σ and Σ ′ are obtained by blowing down some of the exceptional components of the fiber of Z over b 0 . If we set L := Σ * (L) and M := (Σ ′ ) * (M), then (11.3) gives that L π −1 (B\b 0 ) ∼ = M π −1 (B\b 0 ) .
Lemma 10.7 now gives that there exists a Cartier divisor D on Z supported on π −1 (b 0 ) such that gives that d and d ′ are equivalent on Z. Moreover, d is strictly balanced by the above Reduction and d' is strictly balanced by the assumption (11.2) together with Corollary 8.6. Therefore, we can apply Lemma 3.11 twice to conclude that X = Y. Now, the relation (11.3) together with the Lemma 10.7 imply that there exists a Cartier divisor D ′ on X = Y supported on π −1 (b 0 ) such that (11.7) L = M ⊗ O X (D ′ ).

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In particular, we get that d is equivalent to d ′ . Since d and d ′ are strictly balanced, Lemma 3.10 implies that d = d ′ . Since [Y i ′ ֒→ P r × B] b 0 = [Y ֒→ P r ] ∈ Hilb ss d by assumption (11.2), Corollary 10.8 gives that [X ⊂ P r ] ∈ Hilb ss d , q.e.d. The proof of part (2) is similar: it is enough to replace quasi-stable curves by quasip-stable curves (using Corollaty 6.2), to replace the stable reduction by the p-stable reduction and using the fact that the moduli space M ps g of p-stable curves of genus g is separated.
From the above Theorem 11.1, we can deduce a description of the Hilbert and Chow polystable and stable points of Hilb d .
principal GL r+1 -bundle: call it E. To find the G-equivariant morphism to H d , consider the family X E := X × S E of quasi-p-stable curves together with the pullback of L to X E , call it L E , whose restriction to the geometric fibers is properly balanced.
By definition of frame bundle, f E * (L E ) is isomorphic to A r+1 k × k E. Moreover, the line bundle L E is relatively ample by Remark 5.6; hence it is relatively very ample by Theorem 13.5(iii). Therefore, L E gives an embedding over E of X E in P r × E. By the universal property of the Hilbert scheme Hilb d , this family determines a map ψ : E → Hilb d whose image is contained in H d by Theorem 11.1(2). It follows immediately from the construction that ψ is a G-equivariant map. Consider an isomorphism between two pairs (f : X → S, L) and (f ′ : X ′ → S, L ′ ) , i.e., an isomorphism h : X → X ′ over S and an isomorphism of line bundles L ∼ = h * L ′ . Since f ′ h = f , we get a unique isomorphism between the vector bundles f * (L) and f ′ * (L ′ ). As taking the frame bundle gives an equivalence between the category of vector bundles of rank r +1 over S and the category of principal GL r+1bundles over S, the isomorphism f * (L) ∼ = f ′ * (L ′ ) leads to a unique isomorphism between their frame bundles, call them E and E ′ respectively. It is clear that this isomorphism is compatible with the G-equivariant morphisms ψ : E → H d and ψ ′ : E ′ → H d .
Conversely, given a section (φ : E → S, ψ : E → H d ) of [H d /G] over a k-scheme S, let us construct a family of quasi-p-stable curves of genus g over S and a line bundle whose restriction to the geometric fibers is properly balanced of degree d.
Let C d be the restriction to H d of the universal family on Hilb d . By Theorem 11.1(2), the pullback of C d by ψ gives a family C E on E of quasi-p-stable curves of genus g and a line bundle L E on C E whose restriction to the geometric fibers is properly balanced. As ψ is G-invariant and φ is a G-bundle, the family C E descends to a family C S over S, where C S = C E /G. In fact, since C E is flat over E and E is faithfully flat over S, C S is flat over S too. Now, since G = GL r+1 , the action of G on C d is naturally linearized. Therefore, the action of G on E can also be linearized to an action on L E , yielding descent data for L E . Since L E is relatively very ample and φ is a principal G-bundle, a standard descent argument shows that L E descends to a relatively very ample line bundle on C S , call it L S , whose restriction to the geometric fibers of C S → S is properly balanced by construction.
It is straightforward to check that an isomorphism on [H d /G](S) leads to an unique isomorphism in J ps d,g (S). 12.3. Fibers of Φ ps : J ps d,g → M ps g . The aim of this subsection is to give a description of the fiber of Φ ps over a p-stable curve X ∈ M ps g in terms of the Simpson's compactified Jacobian of X, Jac d (X), that we will now describe.
Let X be a p-stable curve of genus g. Let I be a coherent sheaf on X. We say that I is torsion-free if supp(I) = X and I does not have non-zero subsheaves whose support has dimension zero. Clearly, a torsion-free sheaf I can be not free only at the nodes and cusps of X. We say that I is of rank-1 if I is invertible on a dense open subset of X. Each line bundle on X is torsion-free of rank-1.
For each subcurve Y of X, let I Y be the restriction I |Y of I to Y modulo torsion. If I is a torsion-free (resp. rank-1) sheaf on X, so is I Y on Y . We let deg Y (I) denote the degree of I Y , that is, deg Y (I) := χ(I Y ) − χ(O Y ).
Definition 12.9. Let X be a p-stable curve of genus g ≥ 2 and I a rank-1 torsion-free sheaf of degree d on X. We say that I is ω X -semistable if for every proper subcurve Y of X, we have that where k Y denotes, as usual, the length of the subscheme Y ∩ X \ Y of X.
Consider the controvariant functor (12.5) J d,X : SCH → SET which associates to a scheme T the set of T -flat coherent sheaves on X × T which are rank-1 torsion-free sheaves and ω X -semistable on the geometric fibers X × {t} of the second projection morphism X × T → T . From the work on Simpson in [Sim94], we have the following result concerning the co-representability of the moduli functor J d,X (in the sense of Convention 1.8).
Fact 12.10 (Simpson). For any integer d, there is a projective variety Jac d (X) which co-represents the functor J d,X . The geometric points of Jac d (X) parametrize Sequivalence classes of rank-1 torsion-free sheaves on X which are ω X -semistable.
For the definition of S-equivalence of sheaves, we refer the interested reader to [Sim94].
In the next Lemma we will describe torsion-free, rank-1 sheaves on X via certain line bundles on quasi-p-stable models of X.
Lemma 12.11. Let I be a rank-1 torsion-free sheaf on a p-stable curve X. Then there exists a quasi p-stable curve X ′ and a line bundle L on X ′ such that (i) • ps(X ′ ) = X; • deg E L = 1 for all exceptional subcurves E of X ′ ; • I = π * (L) where π : X ′ → ps(X ′ ) = X is the natural morphism. Moreover, X ′ and L | X ′ are unique.

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(ii) Furthermore, I is ω X -semistable if and only if L is properly balanced.
Proof. We start by proving (i). Denote by Sing(I) the set of singular points of X where I is not locally-free. Then the partial normalization of X at Sing(I), ν : Y → X, and the blow-up (in the sense of Corollary 2.11) of X at Sing(I), π : X ′ → X, fit into the following commutative diagram where i represents the natural inclusion morphism. CLAIM 1: There is a unique line bundle M on Y such that ν * (M ) = I. This is certainly well-known (see [Kas] and the references therein), so we only give a sketch of the proof. Consider the sheaf End(I) of endomorphisms of I. Scalar multiplication induces a natural inclusion O X ֒→ End(I) and this inclusion makes End(I) into a sheaf of finite commutative O X -algebras. Moreover, there exists a unique rank-1 torsion-free sheaf J on Spec(End(I)) with the property that f * (J) = I, where f : Spec(End(I)) → X is the natural map (see [Kas,Lemma 3.7]). The claim now follows from the following two facts (*) End(I) = ν * (O Y ) and J is a line bundle.
Indeed, if (*) is true then Y = Spec(End(I)) and we can take M = J. Property (*) is a local property, i.e. it is enough to prove that for any p ∈ X with ν −1 (p) = {q 1 , · · · , q r } ⊂ Y , we have that (**) End(I p ) ∼ = ⊕ i O Y,q i as O X,p − modules, I p is a free module over End(I p ).
If p ∈ Sing(I) then (**) is clear: ν is an isomorphism above p and I p = O X,p is a free module over End(I p ) = O X,p . If p ∈ Sing(I) (hence p is a node or a cusp of X), then it is well-known (see e.g. [Kas,Prop. 5.7]) that I p is isomorphic to the maximal ideal m p of O X,p , End(m p ) is isomorphic to the normalization O X,p of O X,p and m p is a free module over O X,p . Property (**) is proved also in this case, q.e.d. Let now E := E 1 ∪ · · · ∪ E n be the union of the exceptional subcurves of X ′ . Then we can find a line bundle L on X ′ such that L |Y = M and deg E i L = 1, i = 1, . . . , n. The proof of part (i) is now implied by the following CLAIM 2: The natural restriction morphism res : π * L → ν * (L |Y ) = ν * (M ) is an isomorphism of sheaves on X.
We must show that for every open subset U ⊆ X, the restriction map res : L(π −1 (U )) → L |Y (i −1 π −1 (U )) = L |Y (ν −1 (U )) Let now (f : Y → S, L) ∈ P d,X (S) and consider the p-stable reduction of f (see Definition 2.12) | | z z z z z z z z S together with the sheaf π * (L) on ps(Y) whose restriction to the geometric fibers is torsion-free and rank-1. By sending (f : Y → S, L) into (ps(f) : ps(Y) → S, π * (L)), we get a natural transformation of functors P d,X → J d,X . Lemma 12.11 implies that this natural transformation of functors is a local isomorphism, q.e.d. The proof of the Theorem now follows by combining the above three Claims and using the fact (recalled in Convention 1.8) that two locally isomorphic functors are co-represented by the same scheme.