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Mukai models and Borcherds products

Let F_{g,n} be the moduli space of n-pointed K3 surfaces of genus g with at worst rational double points. We establish an isomorphism between the ring of pluricanonical forms on F_{g,n} and the ring of certain orthogonal modular forms, and give applications to the birational type of F_{g,n}. We prove that the Kodaira dimension of F_{g,n} stabilizes to 19 when n is sufficiently large. Then we use explicit Borcherds products to find a lower bound of n where F_{g,n} has nonnegative Kodaira dimension, and compare this with an upper bound where F_{g,n} is unirational or uniruled using Mukai models of K3 surfaces in g<21. This reveals the exact transition point of Kodaira dimension in some g.


Introduction
The moduli space of primitively polarized K3 surfaces of genus g is identified with a Zariski open set of a 19-dimensional modular variety F g = Γ g \D g of orthogonal type, thanks to the Torelli theorem [47], [11]. Its complement is the (−2)-Heegner divisor and parametrizes K3 surfaces having rational double points. Over F g we have the moduli space F g,n of n-pointed K3 surfaces of genus g with at worst rational double points ( §2). The purpose of this paper is to establish a correspondence between pluricanonical forms on F g,n and modular forms on F g , and give applications to the birational type of F g,n . Specifically, we prove that the Kodaira dimension κ(F g,n ) stabilizes to 19 when n is sufficiently large; and for g ≤ 20 we study the transition point of κ(F g,n ) from −∞ to ≥ 0 by combining the modular form method using explicit Borcherds products and the geometric method using Mukai models of polarized K3 surfaces. In the course we will also observe a curious coincidence (1.3) between the Borcherds products and the Mukai models.
Let H 0 (F • g,n , K ⊗m F • g,n ) be the space of m-canonical forms on the regular locus F • g,n of F g,n . Let M k (Γ g , χ) be the space of modular forms on D g of weight k and character χ with respect to Γ g , S k (Γ g , χ) be the subspace of cusp forms, and M k (Γ g , χ) (m) be the subspace of modular forms which vanish to order ≥ m at the (−2)-Heegner divisor. We have M k (Γ g , det) (1) =M k (Γ g , det) (Corollary 3.3). Our point of departure is the following correspondence.
If we consider only pluricanonical forms on the moduli space of smooth n-pointed K3 surfaces, the modular forms can be meromorphic at the (−2)-Heegner divisor. Therefore, in order to fully understand the connection with modular forms, it is necessary to extend the universal family over the whole modular variety by allowing rational double points. (See also Remark 4.10.) As an application of (1.2), we study the birational type of F g,n . By the Kodaira dimension κ(F g,n ) of F g,n , we mean the Kodaira dimension of its smooth projective model X. Then κ(F g,n ) is nondecreasing with respect to n ( [34]), and is bounded by 19 ([32]). The question whether κ(F g,n ) indeed arrives at 19 in large n has remained (cf. [5]). The interest is in the range g ≤ 62, since κ(F g ) = 19 for g ≥ 63 and some other g by Gritsenko-Hulek-Sankaran [23]. The correspondence (1.2) implies the following. Corollary 1.2. Let k 0 be a weight such that S k 0 (Γ g , det) {0}, k 1 be a weight such that dim S k 1 (Γ g , det) > 1, and k 2 be a weight such that S k 2 (Γ g , det) gives a generically finite map F g P N . Then (1) κ(F g,n ) ≥ 0 for all n ≥ k 0 − 19, (2) κ(F g,n ) > 0 for all n ≥ k 1 − 19, and (3) κ(F g,n ) = 19 for all n ≥ k 2 − 19. In particular, κ(F g,n ) stabilizes to 19 for sufficiently large n.
Here (3) holds because the canonical map of a smooth projective model X factors through F g,n ։ F g P N where F g P N is the rational map defined by S 19+n (Γ g , det) (Corollary 4.7), and hence its image has dimension 19 when n = k 2 − 19. The existence of such a weight k 2 is guaranteed by the general theory of Baily-Borel ( [3]) and Hirzebruch-Mumford proportionality ( [24]).
We can always find, for every g, an explicit cusp form F(g) of character det by quasi-pullback of the Borcherds Φ 12 form ( [8], [9], [35], [23]). If k(g) is the weight of F(g) and n(g) = k(g)−19, then F g,n(g) has positive geometric genus, and κ(F g,n ) ≥ 0 for n ≥ n(g). Usually the Borcherds product F(g) has been used only when k(g) ≤ 19 to study the birational type of F g ( [35], [23]). But even when k(g) > 19, it can be thus used to study the universal families over F g .
One observes that n(g) is close to n ′ (g) in relatively many cases. Thus sandwich by Mukai models and Borcherds products, two techniques of different nature, tells us a rather precise information on the transition of the birational type of F g,n . This is what the title means. Moreover, in every 3 ≤ g ≤ 10, we find ( §5.2.8) that the coincidence (1. 3) n(g) = dim V g holds, where V g is a representation of an algebraic group appearing in the classical/Mukai models. This might suggest a further link between the Borcherds product F(g) and the Mukai model. A related subject is the Kodaira dimension of the moduli space M g,n of n-pointed curves of genus g, which has been studied by Logan [36], Farkas [16] and Agostini-Barros [1] (and for n = 0, Harris-Mumford-Eisenbud [29], [14] and Farkas-Jensen-Payne [17]). Incidentally, in many cases in 7 ≤ g ≤ 22, n(g) is near to the Logan-Farkas bound for κ(M g,n ) ≥ 0.
As for the criterion (2) in Corollary 1.2, we can find a weight k 1 such that dim S k 1 (Γ g , det) > 1 by multiplying F(g) and a space M k (Γ g ) = M k (Γ g , 1) with dim M k (Γ g ) > 1. Such a space M k (Γ g ) can be explicitly found, e.g., by the Jacobi lifting [22]. Similarly, for the criterion (3), we can find a weight k 2 there by multiplying F(g) and a space M k (Γ g ) that gives a generically finite map F g P N . Although we know that such a weight k exists by the Baily-Borel theory, it is in general not easy to explicitly calculate it. Moreover, the resulting bound n(g) + k for κ(F g,n ) = 19 would be far from the actual bound, which is expected to be near to n(g).
These points could be improved if F g,n admits a compactification X which is an equidimensional family over some toroidal compactification of F g in codimension 1 ( §4.3). For such a compactification, the boundary obstruction for pluricanonical forms can be estimated in terms of modular forms. Consequently, (1.1) extends to an isomorphism with the log canonical ring of F g,n ֒→ X, and moreover, assuming X has canonical singularities, we have κ(F g,n ) = 19 whenever n > n(g). However, unlike the case of abelian varieties, no example of such a compactification has been known, which makes this part conditional. The singularities of F g,n can be studied through the theory of semi-universal deformation of rational double points and the theory of automorphisms of K3 surfaces. Anyway, Corollary 1.2 enables to study κ(F g,n ) without knowing explicit compactification.
The origin of this paper goes back to the work of Shioda [51] on the correspondence between elliptic cusp forms of weight 3 and canonical forms on the elliptic modular surfaces. This was generalized by Shokurov [52] to n-pointed elliptic curves, and by Hatada [30] to n-pointed abelian varieties and Siegel modular forms. Another generalization of elliptic curves are K3 surfaces. The new feature here is the degenerate family over the (−2)-Heegner divisor, which is responsible for the vanishing condition there. This paper is organized as follows. In §2 we give an analytic construction of F g,n . In §3 we recall the basic theory of orthogonal modular forms. In §4 we prove Theorem 1.1. In §5 we prove Theorem 1.3.
I would like to thank Igor Dolgachev for his help in genus 3, and Gavril Farkas, Shigeru Mukai and Alessandro Verra for their valuable comments. I would also like to thank the referee for many helpful comments.

Universal K3 surface over the period domain
In this section we give an analytic construction of F g,n . This is necessary for the connection with modular forms. The effort is paid for equivariant extension over the (−2)-Heegner divisor. The main results are Propositions 2.1 and 2.13.
Let g ≥ 2. Let Λ K3 = 3U ⊕ 2E 8 be the K3 lattice. We fix a primitive vector l ∈ Λ K3 of norm 2g − 2, which is unique up to the O(Λ K3 )-action ( [45]). The polarized K3 lattice of degree 2g − 2 is defined as its orthogonal The polarized period domain is defined by This consists of two connected components, each of which is a Hermitian symmetric domain of type IV. LetÕ(Λ g ) be the kernel of the reduction map O(Λ g ) → O(Λ ∨ g /Λ g ). By Nikulin [45],Õ(Λ g ) is identified with the stabilizer of l in O(Λ K3 ). We set SinceÕ(Λ g ) has an element exchanging the two components of Ω g , F g is irreducible. By Baily-Borel [3], F g is a normal quasi-projective variety.
The (−2)-Heegner divisor of Ω g is defined as This is locally finite and descends to an algebraic divisor of F g . We write A marked polarized K3 surface of genus g is a triplet (S , L, ϕ) where S is a smooth K3 surface, L is an ample line bundle on S of degree 2g − 2, and ϕ is an isometry H 2 (S , Z) → Λ K3 such that ϕ([L]) = l. (In particular, L is assumed to be primitive.) When L is nef and big, we say instead quasipolarized. By the period mapping, Ω • g is identified with the fine moduli space of marked polarized K3 surfaces of genus g. Gluing the polarized Kuranishi families, we have a universal family (X • g → Ω • g , ϕ) over Ω • g (see [6], [31]). We have the equivariant action ofÕ(Λ g ) on X • g → Ω • g induced by changing the marking (cf. [31] p.120).
Our purpose in this section is to extend X • g → Ω • g to a family over Ω g together with the action ofÕ(Λ g ), and determine its fixed divisor. The results proved in §2.2 and §2.3, after preliminaries in §2.1, are summarized as follows.
(1) The family X • g → Ω • g extends to a projective flat family π : X g → Ω g over Ω g of polarized K3 surfaces with at worst rational double points, and theÕ(Λ g )-action on X • g extends to aÕ(Λ g )-action on X g . (2) The reflection with respect to a (−2)-vector δ ∈ Λ g acts trivially on the total space π −1 (δ ⊥ ∩Ω g ) over δ ⊥ ∩Ω g . When g ≥ 3, every fixed divisor of theÕ(Λ g )-action on X g is of this form. When g = 2, we have an additional fixed divisor formed by the ramification curves of the double planes.
In §2.4 we take the n-fold fiber product X g,n of X g → Ω g and the quotient F g,n =Õ(Λ g )\X g,n . This is an analytic construction of the moduli space of n-pointed K3 surfaces of genus g with at worst rational double points, and will be a basis of the connection with modular forms.

Burns-Rapoport period domain.
Let Let ∆(ω) be the set of (−2)-vectors in NS (ω) and W(ω) be its Weyl group. Connected components of V(ω) − ∪ δ∈∆(ω) δ ⊥ are called Weyl chambers of V(ω). The group W(ω) × {±id} acts freely and transitively on the set of Weyl chambers of V(ω). The Burns-Rapoport period domain ( [11]) is set-theoretically defined as If (S , ϕ) is a marked K3 surface, its Burns-Rapoport period is defined by where K S is the Kähler chamber of S . By the strong Torelli theorem, this identifiesΩ K3 with the fine moduli space of marked K3 surfaces ( [11], [6]). The marked Kuranishi families equipΩ K3 with the structure of a non-Haussdorff complex manifold and a universal family (X K3 π →Ω K3 , ϕ), where ϕ is an isomorphism R 2 π * Z → Λ K3 of local systems. The projectioñ Ω K3 → Ω K3 is locally isomorphic by the local Torelli theorem.
The group O(Λ K3 ) acts onX K3 →Ω K3 equivariantly as follows. Let The isomorphism between the fibers over them is "the identity map" S → S of S . To be more intrinsic, if (S ′ , ϕ ′ ) is a marked K3 surface isomorphic to (S , γ • ϕ), the fiber map is the isomorphism f : ∈ Ω g and that the Weyl chamber C contains l in its closure.
By the Burns-Rapoport period mapping,Ω g is identified with the set of isomorphism classes of marked quasi-polarized K3 surfaces of genus g. The basic observation is This is a consequence of the following general property. Lemma 2.3. Let X → U be a family of K3 surfaces with a line bundle L on X. Let L u = L| X u for u ∈ U. Assume that L u 0 is nef and big for a point u 0 ∈ U. Then L u is nef and big for all u ∈ U in an open neighborhood of u 0 .
Indeed, for a marked Kuranishi family (X → U, ϕ) such that P BR (X u 0 , ϕ u 0 ) ∈Ω g , Lemma 2.3 implies that ϕ −1 u (l) ∈ NS (X u ) is nef and big for all u ∈ P −1 (Ω g ) in a neighborhood of u 0 . Hence, shrinking U if necessary, we have P −1 is an open immersion and Ω g is a complex submanifold of Ω K3 , this shows that Recalling that the complex structure ofΩ K3 is induced from the Kuranishi families, this proves Proposition 2.2. Lemma 2.3 follows from the following cohomological characterization of nef and big line bundle and the upper semicontinuity of h i (mL u ). Proof. The "only if" part is the Kodaira-Ramanujam vanishing theorem for mL. We prove the "if" part. Suppose h 1 (mL) = h 2 (mL) = 0. Since h 0 (mL) > 0 by Riemann-Roch, L is in the positive cone of NS (S ) R . Assume to the contrary that L is not nef and big. Then L is not nef, i.e., not in the closure of the ample cone, because a nef bundle L with (L, L) > 0 is also big. Take a Weyl chamber C of NS (S ) R with L ∈ C. There exists a sequence of effective (−2)-vectors δ 1 , · · · , δ N ∈ NS (S ) such that, if s i is the reflection by δ i and C i = s i (C i−1 ) with C 0 = C, then C N is the ample cone and (C i−1 , δ i ) < 0 for all i. (Connect C and the ample cone by a general segment.) We set L 0 = L and for every i. We look at i = N. Note that (L N , δ k ) ≥ 0 for every k because L N is nef. Let j be the largest index such that (L N , δ j ) > 0, which exists because L L N . There exists an irreducible curve C ≤ δ j with (L N , C) > 0. By Saint-Donat's result ( [49] §2 - §3, see also [31] Chapter 2.3), we may assume that C is smooth. Consider the exact sequence We have h 1 (mL N ) = 0 because L N is nef and big, and h 0 ((mL N + C)| C ) > 0 because deg((mL N + C)| C ) ≥ 2g C by m ≥ 2. It follows that This proves h 0 (mL N ) < h 0 (mL).
On the other hand, since (L N , L N ) = (L, L), this contradicts the consequence of the Riemann-Roch formula This proves Proposition 2.4 and so finishes the proof of Lemma 2.3 and Proposition 2.2.
The fibers of the projection p :Ω g → Ω g are described as follows. For [ω] ∈ Ω g we let NS (ω, l) = l ⊥ ∩ NS (ω) and ∆(ω, l) be the set of (−2)vectors in NS (ω, l). Since NS (ω, l) is negative-definite, the root lattice R(ω, l) ⊂ NS (ω, l) generated by ∆(ω, l) is an orthogonal sum of some ADE root lattices. Let W(ω, l) be its Weyl group. This shows that F g ≃Õ(Λ g )\Ω g , so F g is the moduli space of quasipolarized K3 surfaces of genus g. Equivalently, contracting the (−2)-curves orthogonal to the quasi-polarization, F g is also the moduli space of polarized K3 surfaces of genus g with at worst rational double points.

2.2.
Extension of the universal family. We now construct a family X g → Ω g as in Proposition 2.1 (1). We do this in two steps: first contractX g over Ω g , and then show that this descends to a family over Ω g .
Let (X g π →Ω g , L, ϕ) be the universal family as in §2.1.3. Since L ⊗3 u is base point free for every u ∈Ω g ([49] Theorem 8.3, see also [31] Chapter 2 Remark 3.4), we can take the relative projective morphismX g → P(π * L ⊗3 ) ∨ overΩ g . LetX g be the image of this morphism. This is a flat projective family of K3 surfaces with at worst rational double points. By [28] Corollaire 6.5.4,X g is normal. At each fiber, (X g ) u → (X g ) u contracts all (−2)-curves orthogonal to L u . Proposition 2.6. There exists a family X g → Ω g of K3 surfaces with at worst rational double points such thatX g is isomorphic to the pullback of X g toΩ g .
Proof. Let (X → U, L, ϕ) be a marked quasi-polarized Kuranishi family of (S , L, ϕ) = (X u 0 , L u 0 , ϕ u 0 ). We set [ω] = P(u 0 ) and U • = U − P −1 (H). We may assume that [ω] ∈ H. By Lemma 2.5, a point in the same fiber of Ω g → Ω g as (S , L, ϕ) can be written as ( Outside H the two marked families are isomorphic: the gluing is given by the equivariant action of w −1 on X| U • , which we denote by f : It suffices to show that the closure of the graph of f in X× U X ′ is an analytic set, and that its fiber over u 0 , as a correspondence between S and S ′ , gives an isomorphism between the complements of the exceptional divisors of S →S and S ′ →S ′ . This would imply that f extends to an isomorphism X\Sing(X) →X ′ \Sing(X ′ ) whereX,X ′ are the relative contractions of X, X ′ respectively, which then extends to an isomorphismX →X ′ by normality.
In order to prove our claim, we use a variant of the argument of Burns-Rapoport [11]. For each u ∈ U • , f u : . By the same argument as [11] p.248, the volume of the graph of f u is bounded. This shows that the graph of f has finite volume, hence its closure is an analytic subset of X × U X ′ by [7] Theorem 3. The limit cycle . By the same argument as [11] p.248 -p.250, Γ can be written as We use a variant of the argument of Matsusaka-Mumford in the proof of Theorem 2 of [38].
Let U be the open set of |3L| consisting of members D ∈ |3L| such that D C k for every k.
We suppose, to the contrary, that (C i , L) > 0 for some i. For a general point p of C i we set U p = U ∩|3L− p|. Then U p is non-empty and of codimension Noticing that p is any general point of C i , we see that U p ∩ U q is open dense in both U p and U q for two general points p, q of C i . This contradicts that 3L separates two general points of C i . Hence (C i , L) = 0 for every i. The assertion (C ′ j , L ′ ) = 0 can be proved similarly.
By construction, the isomorphism g : S → S ′ in (2.3) is the minimal resolution of the gluing isomorphismS →S ′ . As will be shown later (Corollary 2.9), this coincides with the equivariant action by w.
Lemma 2.7. The action ofÕ(Λ g ) onX g →Ω g descends to an equivariant action on X g → Ω g .
be two points in the fiber ofΩ g → Ω g over [ω] where w ∈ W(ω, l), and g : S 1 → S 2 be the resolution of the gluing isomorphism betweenS 1 andS 2 .
. What has to be checked is that coincides with the resolution g ′ of the gluing isomorphism betweenS ′ 1 and Recall that graph(g) and graph(g ′ ) are the main components of the limit cycles lim u graph( f u ) and We study the action of the (−2)-reflections.
Proof. It suffices to show that s δ acts trivially on the fiber over a general point [ω] of H δ ; then s δ does so for all [ω] ∈ H δ by continuity. When [ω] is general, we have ∆(ω, l) = {±δ} and W(ω, l) = {id, s δ }, so the fiber of Ω g → Ω g over [ω] consists of two points. Let (S , ϕ) and (S ′ , ϕ ′ ) = (S , s δ •ϕ) be the corresponding marked K3 surfaces. What has to be shown is that the resolution g : S → S ′ of the gluing isomorphismS →S ′ coincides with the equivariant action by s δ . After the original identification S ′ = S , the action of s δ is the identity of S . So it suffices to show that g : S → S ′ = S is also the identity of S . We may assume that for some a ≥ 0. On the other hand, by (2.2), the action of Γ on H 2 (S , Z) is the reflection ϕ −1 • s δ • ϕ with respect to E. Comparing this with (2.4), we see that by this action, we obtain g(E) ∼ (2a − 1)E, so a = 1. Hence g * = id. By the strong Torelli theorem, g = id.
Remark 2.10. When g > 2, the divisor H δ parametrizes a family of latticepolarized K3 surfaces whose very general Picard lattice is ZL ⊕ ZE ≃ 2g − 2 ⊕ −2 or its overlattice of index 2. Proposition 2.8 for g > 2 also follows from the fact that no K3 surface with such a Picard lattice has an involution which acts trivially on the Picard lattice ( [46]).
We can now determine the fixed divisor of theÕ(Λ g )-action on X g . We need to separate the case g = 2, where general members are double planes branched along sextics.
we have that, either γ = s δ for a (−2)-vector δ ∈ Λ 2 , or γ = −id which acts as the covering transformation of the double planes.
Proof. (1) Let g > 2 and γ id, s δ for any (−2)-vector δ. We assume to the contrary that γ fixes an irreducible divisor D of X g . Since g > 2,Õ(Λ g ) does not contain −id, so γ acts nontrivially on Ω g . Then D = X g | B for an irreducible divisor B of Ω g fixed by γ. Since B is not contained in H δ for any (−2)-vector δ, a general fiber S of D → B is nonsingular, equipped with a marking ϕ. Then the equivariant action of γ on S is the identity of S and satisfies ϕ • id = γ • ϕ. This is absurd.
When g > 2,Õ(Λ g ) also contains −reflections by splitting (2−2g)-vectors which fixes a divisor of Ω g (cf. [23]). But they act nontrivially on the smooth fibers over the fixed divisor, so does not fix a divisor of X g .
Before moving on, we also study the relation with the automorphism group. For [ω] ∈ Ω g let G(ω, l) be the stabilizer of [ω] inÕ(Λ g ).
We have a split exact sequence where π is induced from the equivariant action of G(ω, l) on X g . A splitting of (2.5) is obtained for each choice of a point ofΩ g over [ω].
Proof. By Corollary 2.9, W(ω, l) acts trivially on S . We take a point ([ω], C) ofΩ g lying over [ω], and let (S ,L, ϕ) be the corresponding marked quasipolarized K3 surface. ThenS is the minimal resolution of S andL is the pullback of L. Then Aut(S , L) is isomorphic to Aut(S ,L), which in turn is isomorphic to the stabilizer of ([ω], C) inÕ(Λ g ) by the strong Torelli theorem. The last group is the stabilizer of the chamber C in G(ω, l). This defines a section Aut(S , L) ֒→ G(ω, l) of π. In view of Lemma 2.5, this also shows that (2.5) is exact at the middle.
2.4. F g,n . Let n ≥ 1. We take the n-fold fiber product X g,n = X g × Ω g · · · × Ω g X g and its quotient F g,n =Õ(Λ g )\X g,n . SinceÕ(Λ g ) acts properly discontinuously on Ω g , it does so on X g,n . Thus F g,n is an irreducible complex analytic space fibered over F g . Since X g,n is a flat family of normal varieties over the smooth base Ω g , X g,n is normal by [28] Corollaire 6.5.4. Therefore F g,n is also normal.
If (S , L) is the (possibly singular) polarized K3 surface over [ω] ∈ Ω g , the fiber of F g,n → F g over [ω] ∈ F g is S n /Aut(S , L) by Proposition 2.12. Thus F g,n is the moduli space of n-pointed polarized K3 surfaces of genus g with at worst rational double points. Proposition 2.13. Let π : X g,n → Ω g be the projection.
Proof. The ramification divisors are determined by Proposition 2.11. Classification of the irreducible components of the branch divisor reduces to that ofÕ(Λ g )-equivalence classes of (−2)-vectors in Λ g , which is given in [25] Proposition 2.4 (ii).
As our construction is done over the period domain (which is necessary for the connection with modular forms), F g,n is a priori just complex analytic. But actually Proposition 2.14. F g,n is quasi-projective.
Proof. We use the Hilbert scheme construction. Since the basic line of argument is similar to [53] Chapter 7 and [31] Chapter 5 (whereÕ(Λ g )\Ω • g is considered), we will be brief.
Let H be the Hilbert scheme of polarized K3 surfaces (S , L) of degree 2g − 2 with at worst rational double points, projectively embedded by L ⊗3 into P N where N = 18g − 17. The first step is to show that H is nonsingular. In the locus of smooth K3 surfaces, this is proved in [31] Chapter 5.3.1. In the ADE locus, what has to be shown is that the obstruction space Although S is singular, its singularities are complete intersections, so I S /I 2 S is locally free and sits in the short exact sequence 16.17. Taking the Ext(·, O S ) long exact sequence, we obtain the exact sequence . By the same calculation as in [31] p.92, we see that The next step is to apply the argument of [53] §7.3 II. By construction we have a flat projective family X → H of K3 surfaces with at worst rational double points, acted on by G = PGL N+1 . According to loc. cit., with the smoothness of H, there exists a finite group Γ, quasi-projective varieties V, Z with Γ × G acting on V, and morphisms V → H, V → Z, such that V → H is the quotient by Γ and V → Z is a principal G-bundle. Furthermore, there exists a geometric quotient X ′ = (X × H V)/G with a projective morphism X ′ → Z. In particular, the n-fold fiber product X ′ n = X ′ × Z · · · × Z X ′ is quasiprojective. By [53] Corollary 3.46.2, a geometric quotient X ′ n /Γ exists as a quasi-projective variety. The variety X ′ n /Γ is also a geometric quotient of X × H · · · × H X by G, which in turn is naturally biholomorphic to F g,n . Therefore F g,n is quasi-projective.

Modular forms
In this section, we let Λ be a general integral lattice of signature (2, b) with b ≥ 3. (We will specialize to Λ = Λ g in the next section.) We recall the basic theory of modular forms for the orthogonal group of Λ. The material should be well-known, but since we could not find a suitable reference at some points, we tried to be rather self-contained. The results needed in §4 are Corollary 3.3 and Proposition 3.5.  K D ≃ λ ⊗b ⊗ det (cf. [23]). This is a consequence of the Euler sequence for PΛ C and the adjunction formula for the isotropic quadric.
A Γ-invariant holomorphic section of the line bundle λ ⊗k ⊗ χ on D is called a modular form of weight k and character χ with respect to Γ. We denote by M k (Γ, χ) the space of such modular forms. As in (2.1), the (−2)-Heegner divisor, now restricted to the component D, is defined by For a natural number m we denote by M k (Γ, χ) (m) ⊂ M k (Γ, χ) the subspace of modular forms which vanish to order ≥ m along every irreducible component of H. When Γ is torsion-free, this is identified with where λ, χ in the right hand side stand for the descends of the linearized line bundles λ, χ to Γ\D, and H is the image of H in Γ\D. This is a straightforward generalization of the calculation in [23] Lemma 5.2 and [27] p.420. (It seems that in [27] p.420 one has to assume a ∈ 2Z when −σ is a reflection.) LetÕ Thus, for Γ =Õ + (Λ) and χ = det, vanishing at H is automatic. For general Γ and χ = det, vanishing at the ramification divisor of D → Γ\D is automatic when k ≡ b mod 2.
Next we explain the Petersson metric and the invariant volume form. The Hermitian form (·,·) on Λ C defines a Hermitian metric on the line bundle λ, whose positivity follows from the definition of D. By construction this is O + (Λ R )-invariant. On the other hand, since the character χ is unitary, we also have a Γ-invariant Hermitian metric on the corresponding line bundle. These define a Γ-invariant Hermitian metric on the line bundle λ ⊗k ⊗χ which we call the (pointwise) Petersson metric on λ ⊗k ⊗ χ.
By the isomorphism (3.1), the Petersson metric on λ ⊗b ⊗ det defines an O + (Λ R )-invariant Hermitian metric on K D which we denote by ( , ) K . We define a volume form vol D on D by where Ω is an arbitrary nonzero vector of (K D ) [ω] . This does not depend on the choice of Ω. Since ( , 3.2. Tube domain realization. We recall, in some detail, tube domain realization of D and Fourier expansion of modular forms. This is necessary for the proof of Propositions 3.5 and 4.11. Let I be a rank 1 primitive isotropic sublattice of Λ. This corresponds to a 0-dimensional rational boundary component of D (cf. [3], [50]). We put Λ(I) = I ⊗ (I ⊥ /I). The quadratic form on I ⊥ /I and an isomorphism I ≃ Z (unique up to ±1) define a canonical quadratic form on Λ(I). We write Γ(I) Q for the stabilizer of I Q in O + (Λ Q ). The unipotent radical U(I) Q of Γ(I) Q consists of the Eichler transvections (see [50], [26]) where l ∈ I Q and m ∈ I ⊥ Q . This induces a canonical isomorphism where m ∈ I ⊥ Q is a lift ofm ∈ (I ⊥ /I) Q . The projection PΛ C P(Λ/I) C from PI C ∈ PΛ C defines an embedding We also choose a nonzero vector w 0 in the representation line of χ. Then modular forms F = f (s ⊗k l ⊗ w 0 ) of weight k and character χ for Γ are identified with holomorphic functions f on D satisfying [ω] ∈ D, γ ∈ Γ. Over D I the Petersson metric and the invariant volume form are expressed as follows (compare with [10] p.96 for (2)).
(2) We take coordinates z 1 , · · · , z b on Λ(I) C . By the isomorphism (3.1), the flat canonical form Ω = dz 1 ∧ · · · ∧ dz b corresponds to s ⊗b l ⊗ w 0 up to a constant because both are U(I) C   Proof. This should be a standard fact: see, e.g., [20] Chapter III, §2 for a similar result in the case of Sp(2g, Z) and weight g + 1. But since we could not find a suitable reference for the present case, we give an outline of the proof for the sake of completeness.
It is convenient to work with a toroidal compactification ( [2]). The input for constructing a toroidal compactification of Γ\D is a collection Σ = (Σ I ) I of Γ(I) Z -admissible cone decomposition Σ I of C + I , one for each Γequivalence class of rank 1 primitive isotropic sublattices I (independently). We can take Σ I to be regular with respect to U(I) Z . Let T I = U(I) Z \D. Via ι I ′ , T I is realized as an open set of the torus U(I) Z \U(I) C , and the fan Σ I defines a toroidal embedding T I ֒→ T Σ I . This is the partial compactification in the direction of the 0-dimensional cusp I. The domain D has also 1-dimensional cusps, but the partial compactification there is canonical and has etale glueing maps to the partial compactifications at the adjacent 0dimensional cusps I. (This corresponds to a boundary ray in C + I .) The toroidal compactification (Γ\D) Σ of Γ\D is obtained by glueing Γ\D and quotients of the partial compactifications, essentially along small neighborhoods of the boundary. For a cone σ ∈ Σ I , we write T σ I ⊂ T Σ I for the toroidal embedding by σ, and let U = U x ⊂ T σ I be a small neighborhood of an arbitrary point x of the (snc) boundary divisor of T σ as |q| → 0, where q = re iθ and vol ′ I is a flat volume form on the boundary divisor. Since k ≥ b, this shows that U (F, F) k,χ vol D < ∞ if and only if f (Z) → 0 as |q| → 0. In the Fourier expansion (3.4), this is equivalent to a m = 0 for all m with (m, v 0 ) = 0. When (v 0 , v 0 ) > 0, this is equivalent to , so this is equivalent to a m = 0 for all m ∈ σ ∩ U(I) ∨ Z . Since every boundary ray of C + I is also a ray in Σ I , this implies the above equivalence and so the "if " part of the proposition.
Conversely, if F is a cusp form, we can check U (F, F) k,χ vol D < ∞ for U with dim σ general by a similar asymptotic estimate.

Pluricanonical forms and modular forms
In this section we prove Theorem 1.1 based on the results of §2 and §3. The proof will be completed at §4.2. In §4.3 we discuss extension over a certain compactification of F g,n , assuming some properties.
For the consistency with §3, we need to restrict attention to a connected component. We choose a component D g of Ω g and put Γ g = O(Λ g ) ∩ O + (Λ g ). Let X + g,n be the connected component of X g,n over D g and π : X + g,n → D g be the projection. SinceÕ(Λ g ) contains an element exchanging the two components of Ω g , we have F g,n = Γ g \X + g,n . We write λ for the restriction of the tautological bundle O(−1) to D g (or to Ω g ).

4.1.
Relative canonical forms and modular forms. First we clarify the relation between relative canonical forms on X + g,n and modular forms. The main attention will be on the degenerate family over H. We begin with the following remark.
Lemma 4.1. The canonical divisor K X g,n of X g,n is Cartier.
Proof. Let (X → U, L) be a quasi-polarized Kuranishi family,π : X n → U be its n-fold fiber product, andX n → U be its relative contraction as before. It suffices to show that KX n is Cartier. Since Kπ| F ≃ O F for each fiber F of π, we have Kπ ≃ O X n after shrinking U, so K X n ≃ O X n . This shows that K (X n ) reg ≃ O (X n ) reg . SinceX n is normal and its singular locus has codimension ≥ 3, K (X n ) reg extends to an invertible sheaf isomorphic to OX n .
Let K π = K X + g,n ⊗ π * K −1 D g be the relative canonical bundle of π. Since K X + g,n and K D g are Γ g -linearized, K π is also Γ g -linearized.
Proposition 4.2. We have Γ g -equivariant isomorphisms (4.1) π * K π ≃ λ ⊗n , K π ≃ π * λ ⊗n of line bundles on D g and on X + g,n . Proof. We argue for π : X g,n → Ω g . Since K π |F ≃ KF ≃ OF for each fiberF of X g,n → Ω g , π * K π is invertible and the natural homomorphism π * π * K π → K π is an isomorphism. Hence the two isomorphisms in (4.1) are equivalent to each other. Furthermore, we can use induction on n to reduce the problem to the case n = 1. Indeed, in the cartesian diagram X g,n So we let n = 1 and prove π * K π ≃ λ. Let (X gπ →Ω g , ϕ) be the smooth universal family overΩ g as in §2.1.3 andπ :X g →Ω g be its contraction. The projection p :Ω g → Ω g is the period map for the family (X g →Ω g , ϕ). For each u ∈Ω g , writing S = (X g ) u , the marking ϕ u sends H 2,0 (S ) ⊂ H 2 (S , C) to the line in (Λ K3 ) C corresponding to p(u) ∈ Ω g . This means thatπ * Kπ ≃ p * λ. Sinceπ * Kπ ≃π * Kπ, we obtainπ * Kπ ≃ p * λ overΩ g . We want to show that this descends to π * K π ≃ λ over Ω g . At each u ∈Ω g , writingS = (X g ) u , the isomorphismπ * Kπ → p * λ is given by Even if we use another marking w • ϕ u where w ∈ W(p(u), l), this composition map does not change because w acts trivially on the line ϕ u (H 2,0 (S )). Henceπ * Kπ ≃ p * λ descends to π * K π ≃ λ. By construction, this isÕ(Λ g )equivariant.
The Petersson metric on λ ⊗n is the L 2 metric on π * K π . Proposition 4.3. Let F be a section of λ ⊗n over a subset U of D g . Let Ω be the relative canonical form on π −1 (U) ⊂ X + g,n corresponding to F. Then where π −1 (U)/U means the fiber integral.
Proof. This is a pointwise assertion. Let [ω] ∈ D g and (S , L, ϕ) be a corresponding marked quasi-polarized K3 surface with the contraction µ : S → S . We first consider the case n = 1.

4.2.
Proof of Theorem 1.1. Now we prove Theorem 1.1. Let (g, n) (2, 1). First we derive the isomorphism (1.1). As an intermediate step, we choose a torsion-free normal subgroup Γ ⊳ Γ g of finite index and put F ′ g,n = Γ\X + g,n and F ′ g = Γ\D g . Let π : F ′ g,n → F ′ g be the projection, H ′ ⊂ F ′ g be the image of H, and B ′ = π * H ′ ⊂ F ′ g,n be the total space over H ′ . Lemma 4.4. For each m, l ≥ 0 we have . Proof. By (3.1) and Proposition 4.2, we have Γ g -equivariant isomorphisms ) of line bundles on X + g,n . This descends to an isomorphism K ⊗m Taking the global sections over F ′ g,n , we obtain (4.3).
Let G = Γ g /Γ. We take the G-invariant part of the isomorphism (4.3) with l = m. For the right hand side, we have by definition. For the left hand side, let p : F ′ g,n → F g,n be the quotient map by G and F • g,n be the regular locus of F g,n . Note that the singular locus of F g,n is of codimension ≥ 2 by the normality of F g,n . By the first statement of Proposition 2.13 (1), the composition is doubly ramified at π −1 (H) and has no other ramification divisor. Since X + g,n → F ′ g,n is unramified, we find that the ramification divisor of p is B ′ with ramification index 2. This shows that p * K F • g,n ≃ K F ′ g,n (−B ′ ) over p −1 (F • g,n ) ⊂ F ′ g,n by the Hurwitz formula. Therefore we obtain . This proves the isomorphism (1.1). Compatibility of multiplication is evident.
Next we derive the isomorphism (1.2). We use the following equality. Proof. Since the problem is local, we may assume that U is a small open subset of D g , F is a section of λ ⊗19+n ⊗ det over U, and Ω is the canonical form on π −1 (U) ⊂ X + g,n corresponding to π * F by the isomorphism K X + g,n ≃ π * (λ ⊗19+n ⊗det). Since U is small, we can decompose F as F = F 1 ⊗ F 2 with F 1 a section of λ ⊗19 ⊗ det and F 2 a section of λ ⊗n . Let Ω 1 be the canonical form on U corresponding to F 1 , and Ω 2 be the relative canonical form on π −1 (U) corresponding to F 2 . Then Ω = π * Ω 1 ∧Ω 2 . By (3.2) and Proposition 4.3, we see that This proves Lemma 4.5.
The isomorphism (1.2) now follows by comparing, via Lemma 4.5, Proposition 3.5 with the well-known fact that a canonical form Ω on a Zariski open set X • of a smooth proper variety X extends holomorphically over X if and only if X • Ω ∧Ω < ∞. The proof of Theorem 1.1 is completed.
Example 4.6. Let g = 2. When k is even, we have M k (Γ 2 , det) = {0} because −id ∈ Γ 2 acts by −1 on λ ⊗k ⊗ det. Therefore, when both n and m are odd, there is no nonzero m-canonical form on F 2,n nor its smooth projective model.
Explicitly, the correspondence (1.2) is given as follows. We take s l , w 0 as in §3.2. Let Ω 1 be the canonical form on D g corresponding to s ⊗19 l ⊗ w 0 . (This is the flat canonical form dz 1 ∧· · ·∧dz 19 in the tube domain realization at Zl.) Let Ω 2 be the relative canonical form on X + g,1 corresponding to s l . If F = f (s ⊗19+n l ⊗ w 0 ) is a cusp form and Ω F is the corresponding canonical form on X, the pullback of Ω F by the projection p : X + g,n → F g,n is (4.4) p * Ω F = (π * f ) π * Ω 1 ∧ π * 1 Ω 2 ∧ · · · ∧ π * n Ω 2 , where π i : X + g,n → X + g,1 is the i-th projection. This description implies the following.
Corollary 4.7. The canonical map of X factors through F g,n ։ F g P N where F g P N is the rational map associated to S 19+n (Γ g , det).
Corollary 4.8. Let Y be a smooth projective model of F g,n /S n . Then (1.2) also induces an isomorphism S 19+n (Γ g , det) ≃ H 0 (K Y ).
Proof. By (4.4), Ω F is S n -invariant and hence descends to a canonical form on an open set of Y. Since it has finite L 2 norm, it extends over Y.
When g > 2, Y can be thought of as a smooth projective model of the universal family of the Hilbert schemes Hilb n (S ).
The divisor of Ω F on F g,n is described as follows. Let div(F) ′ be the divisor of F over F g , where the vanishing order at a branch divisor of D g → F g is counted as 1/2 of the vanishing order at the ramification divisor over it. Let H be the image of H in F g and π : F g,n → F g be the projection.
Proof. This follows from (4.4) and the Hurwitz formula, noticing that π * H is reduced and π * H ′ has multiplicity 2 for other branch divisors H ′ .
Remark 4.10. If one is interested only in (1.2), one can in fact dispense with the partial compactification F g,n , because Γ\D (F, F) k,χ vol D must diverge if a (meromorphic) modular form F has pole at H. However, in order to understand a full picture of the connection of modular forms and the geometry of universal family, such as pluricanonical forms and Eichler-Shimura theory, it is necessary to fill the (−2)-Heegner divisor.

Extension over compactification.
In this subsection we assume that X is a complex analytic variety containing F g,n as a Zariski open set and regular in codimension 1, such that F g,n → F g extends to a morphism π : X → F Σ g to some toroidal compactification F Σ g of F g with the property that every irreducible component of the boundary divisor ∆ X of X dominates some irreducible component of the boundary divisor ∆ F of F Σ g . We explain how the results of Theorem 1.1 could be extended for such X, clarifying the contribution from the boundary ∆ X . This assumption means that except some codimension ≥ 2 locus X is an equidimensional family over F Σ g , but at present no example of such a compactification has been known, which makes the next assertion conditional. Proposition 4.11. Let X be a complex analytic variety as above.
(1) The isomorphism (1.1) extends to This maps S (19+n)m (Γ g , det m ) (m) into H 0 (X, K ⊗m X ((m − 1)∆ X )). (2) Assume moreover that X is compact and has canonical singularities. Then where the right hand side is the Iitaka dimension of the Q-divisor (19+n)λ− ∆ F − H/2 of F Σ g . Proof. Since this is similar to the case of Siegel modular forms [37], we will just indicate the outline. Let ∆ be a component of ∆ X , and ∆ ′ be a boundary divisor of a torus embedding which gives the component π(∆) of ∆ F . We take a small neighborhood ∆ x ⊂ ∆ of a general point x of ∆ and let ∆ ′ y ⊂ ∆ ′ be a small open set whose image is π(∆ x ). Let U ε , U ′ ε be annulus bundles around ∆ x , ∆ ′ y respectively, say of radius [ε, 1]. Let F be a local modular form of weight k = (19 + n)m and character χ = det m on a neighborhood of π(∆ x ), and Ω be the m-canonical form corresponding to F. Both (1) and (2) boil down to the assertion that Ω has at most pole of order m along ∆ x .
By the L 2/m -criterion in [37] §6, it is sufficient to show that U ε ||Ω|| 2/m = o(ε −α ) for every α > 0. As before, we can relate the L 2/m norm of Ω to the Petersson norm of F. Pulling back to the torus embedding, the problem is reduced to showing that for every α > 0. This can be proved as in the proof of Proposition 3.5: When F is a cusp form, the integral U ′ ε (F, F) 1/m k,χ vol D converges and hence U ε ||Ω|| 2/m converges. This proves the second assertion in (1).

Mukai models and quasi-pullback of Φ 12
In this section we prove Theorem 1.3. In §5.1 we compute the weight k(g) of the quasi-pullback F(g) of the Borcherds Φ 12 form for each g ≤ 22. By Corollary 1.2, we have κ(F g,n ) ≥ 0 for all n ≥ n(g) where n(g) = k(g) − 19. In §5.2 we proceed from the opposite direction: for many g ≤ 20, we study an upper bound n ′ (g) where F g,n is unirational or uniruled by using classical and Mukai models of polarized K3 surfaces.
With the geometric bound n ′ (g), the isomorphism (1.2) can also be used in the opposite direction: we find that S k (Γ g , det) = {0} when k ≤ n ′ (g) + 19.
The closeness of n(g) and n ′ (g) suggests that F(g) would be the cusp form of minimal weight, at least for the character det. 5.1. Quasi-pullback of Φ 12 . Let II 2,26 = 2U ⊕ 3E 8 be the even unimodular lattice of signature (2,26) and D 2,26 be the associated Hermitian symmetric domain. In [8], Borcherds discovered a modular form Φ 12 on D 2,26 of weight 12 and character det for O + (II 2,26 ), whose zero divisor is exactly the (−2)-Heegner divisor of D 2,26 . The quasi-pullback of Φ 12 to D g is defined as follows ( [8], [9]). We choose a primitive vector v g of norm 2 − 2g from E 8 and put K g = v ⊥ g ∩ E 8 . (This is in general not unique even up to O(E 8 ).) This defines a primitive embedding Λ g ֒→ II 2,26 with Λ ⊥ g = K g , and thus an embedding D g ֒→ D 2,26 of domain. We divide Φ 12 by all zeros containing D g and restrict it to D g . More precisely, we put δ (δ, ·) D g where δ ranges over all (−2)-vectors in K g up to ±1, and (δ, ·) is the section of O(1) defined by paring with δ. The basic properties of F(g) are summarized as follows. [35], [23]). Let r(g) be the number of (−2)-vectors in K g . Then F(g) is a modular form on D g of weight k(g) = 12 + r(g)/2 and character det with respect to Γ g , and vanishes at the (−2)-Heegner divisor of D g . When r(g) > 0, F(g) is a cusp form.
As noticed in Corollary 3.3, vanishing at the (−2)-Heegner divisor is in fact automatic. Combining the results so far, we see the following.
Proposition 5.2. Assume that r(g) > 0 for our choice of v g ∈ E 8 . Then a smooth projective model of F g,n(g) has positive geometric genus where n(g) = r(g)/2 − 7. In particular, κ(F g,n ) ≥ 0 for all n ≥ n(g). Furthermore, if we have a compactification of F g,n for some n > n(g) as in Proposition 4.11 (2), then κ(F g,n ′ ) = 19 for all n ′ ≥ n.
Proof. By Proposition 5.1, F(g) is a nonzero element of S k(g) (Γ g , det). By Theorem 1.1, this corresponds to a nonzero canonical form on a smooth projective model of F g,n(g) where n(g) = k(g) − 19 = r(g)/2 − 7. As for the last assertion, when n > n(g), the Q-divisor Then κ(F g,n ) = 19 by Proposition 4.11 (2).
Let Ω F(g) be the canonical form corresponding to F(g). Its divisor on F g,n(g) can be explicitly calculated as follows. For simplicity we assume g 2 mod 4. Let A = Λ ∨ g /Λ g be the discriminant group of Λ g . For λ ∈ A and x ∈ Q <0 , let H(λ, x) = l (l ⊥ ∩ D g ) be the reduced Heegner divisor of discriminant (λ, x) on D g , where l run over all vectors in Λ g + λ ⊂ Λ ∨ g with (l, l) = 2x. In particular, H = H(0, −1). Let H(λ, x) be the (reduced) image of H(λ, x) in F g . On the other hand, for y ∈ Q <0 , let c λ (y) be the number of vectors v in K g + λ ⊂ K ∨ g with (v, v) = 2y, where we identify A ≃ K ∨ g /K g as abelian groups naturally ( [45]).
This can be seen directly as in [9] or using the fact that F(g) is the Borcherds lift of Θ K g (−1) /∆ (see [23] Remark 1). Since H has no common component with H(λ, x − 1) when g 2 mod 4, our assertion follows from Corollary 4.9.
In what follows, we explicitly take a vector v g ∈ E 8 for each g ≤ 22 and calculate n(g) = r(g)/2 − 7. The result is summarized in Theorem 1.3.
Recall ( [12]) that the E 8 -lattice is defined as where we take the (−1)-scaling of the standard quadratic form on Q 8 . The roots of E 8 are as follows. For j k we define δ ± j,±k = (x i ) ∈ E 8 by x j = ±1, x k = ±1 and x i = 0 for i j, k. For a subset S ⊂ {1, · · · , 8} of even cardinality we define δ ′ S = (x i ) ∈ E 8 by x i = 1/2 if i ∈ S and x i = −1/2 if i S . These are the 112 + 128 = 240 roots of E 8 . When g ≤ 8, g 6 and when g = 11, 16, the lattice K g will be obtained by deleting one vertex from the Dynkin diagram of E 8 . We number the vertices of E 8 as follows.
More generally, when 3 ≤ n ≤ 33, F 3,n is rational too. Proof. Consider the (complete) incidence correspondence The fiber of the projection X n → Q n over (p 1 , · · · , p n ) ∈ Q n is the linear system of surfaces in |O Q (3)| passing through p 1 , · · · , p n . Thus, when n ≤ dim |O Q (3)| = 29, X n → Q n is birationally a P 29−n -bundle and hence X n is rational. Therefore F 4,n is unirational. Proof. Consider the incidence correspondence The fiber of the projection X n → (P 5 ) n over general (p 1 , · · · , p n ) ∈ (P 5 ) n is an open dense subset of the sub Grassmannian G(3, V p 1 ,··· ,p n ), where V p 1 ,··· ,p n ⊂ V is the space of quadratic forms vanishing at p 1 , · · · , p n . When n ≤ h 0 (O P 5 (2)) − 3 = 18, we have dim V p 1 ,··· ,p n ≥ 3, so X n is birationally a G(3, 21 − n)-bundle over (P 5 ) n . Therefore More generally, when 7 ≤ n ≤ 17, F 5,n is rational too. 5.2.5. The case g = 6. In [39] §4, Mukai proved that general K3 surfaces of genus 6 are anti-canonical sections of the Fano 3-fold Y of index 2 and degree 5. The variety Y is unique up to isomorphism and is quasihomogeneous for PGL (2). Since h 0 (−K Y ) = 23 and Y is rational, the same argument as before implies Proposition 5.8. F 6,22 is unirational. 5.2.6. The case g = 7. Mukai [39], [43] found two methods of constructing general K3 surfaces of genus 7. Farkas-Verra [18] proved that F 7,n is unirational in n ≤ 8 using the first model ( [39]). Here we use the second model ([43] §3). Let G(2, 5) ⊂ P 9 be the Plücker embedding of the Grassmannian G (2,5), and Y = G(2, 5) ∩ P be the intersection with a general codimension 2 linear subspace P ⊂ P 9 . The variety Y is a quintic del Pezzo 4-fold (unique up to isomorphism), and the stabilizer G ⊂ PGL 5 of Y has dimension 8. Let E be the dual of the universal sub bundle over G (2,5), and put E Y = E ⊗ O(1)| Y . By [43], we have h 0 (E Y ) = 30, and the zero locus of a general section of E Y is a K3 surface of genus 7. This gives a dominant map PH 0 (E Y )/G F 7 whose general fibers are birationally K3 surfaces (Fourier-Mukai partner of the original K3 surface). This construction tells us Proposition 5.9. F 7,14 is uniruled.
The proof shows that the fibration of Fourier-Mukai partners over F 7,14 (dual K3 fibration) is uniruled. This space is akin to F 7,15 , which is the only missing case.
Remark 5.10. The D 5 root system appears in two contexts for g = 7: as a sub root system of K 7 = D 5 ⊕ A 2 , and as the Dynkin diagram of SO(10) whose spinor variety is used in the first Mukai model (cf. [41] (1.11)). 5.2.7. The case 8 ≤ g ≤ 10. Mukai [39] proved that general K3 surfaces of genus 7 ≤ g ≤ 10 are linear sections of a homogeneous space Σ g embedded in PV g for a representation space V g . Using this fact, Farkas-Verra [18] proved that F g,n is unirational in n ≤ g + 1.
5.2.8. Interlude: a coincidence. As we have seen from the classical and Mukai models [39], general K3 surfaces of genus 3 ≤ g ≤ 10 are linear sections of a (quasi-)homogeneous space embedded in PV g for a representation V g of an algebraic group G g . Then we find that the following coincidence always holds: namely, the weight k(g) of F(g) minus 19 (the moduli number) coincides with dim V g . Is this accidental? Is there some deeper link between K3 surfaces and the Borcherds product F(g) behind this coincidence? We summarize (G, V) = (G g , V g ) in the following table.
Here V 6 = S 0 C 2 ⊕ S 8 C 2 ⊕ S 12 C 2 and S + is the half spin representation. In g = 3, 4, 6, 12, n(g) also coincides with h 0 (−K Y ) for Fano 3-folds Y containing the K3 surfaces. This would be a uniform reason why n(g) = n ′ (g) + 1 is attained at these g. 5.2.9. The case g = 11. Barros [4] proved that F 11,n is unirational in n ≤ 6 and uniruled in n = 7. Barros-Mullane [5] proved that κ(F 11,9 ) = 0. Thus the cusp form F (11) gives the unique nonzero canonical form on a smooth projective model of F 11,9 . Farkas-Verra [18] proved that κ(F 11,11 ) = 19. 5.2.10. The case g = 12. Mukai ([41], [43]) proved that general K3 surfaces of genus 12 are anti-canonical sections of Fano 3-folds of genus 12 and Picard number 1. We first recall the construction of such Fano 3-folds following [41] §5. Let Y = G (3,7) and E be the dual of the universal sub bundle over Y. We set V = H 0 ( 2 E) ≃ 2 (C 7 ) ∨ . For a 3-dimensional subspace N of V, we let Y N ⊂ Y be the common zero locus of sections in N. If N is general, then Y N is a Fano 3-fold of genus 12 and Picard number 1. Such Fano 3-folds are rational ([33] Theorem 4.6.7). Restriction of the Plücker embedding Y ֒→ P( 3 C 7 ) gives the anti-canonical embedding Y N ֒→ PN ⊥ ≃ P 13 where N ⊥ ⊂ 3 C 7 is the annihilator of N ⊂ 2 (C 7 ) ∨ . If H is a general hyperplane of PN ⊥ , then S N,H = Y N ∩ H is a K3 surface of genus 12. By Mukai [43], Y N can be uniquely recovered from S N,H inside Y, so that F 12 is birationally a P 13 -bundle over the moduli space G(3, V)/PGL 7 of these Fano 3-folds with fibers the anti-canonical systems.
Proof. Let F ′ 12,13 be the moduli space of 13-pointed Fano 3-folds of genus 12 and Picard number 1. We have the rational map F ′ 12,13 F 12,13 which sends (Y N , p 1 , · · · , p 13 ) to (S N,H , p 1 , · · · , p 13 ) where H = p 1 , · · · , p 13 is the hyperplane of PN ⊥ spanned by p 1 , · · · , p 13 . Since Y N can be recovered from S N,H , we also have the inverse map F 12,13 F ′ 12,13 sending (S N,H , p 1 , · · · , p 13 ) to (Y N , p 1 , · · · , p 13 ). Therefore F 12,13 is birational to F ′ 12,13 . Since F ′ 12,13 is fibered over the unirational base G(3, V)/PGL 7 with general fibers rational (self products of the Fano 3-folds), it is rationally connected by [21]. 5.2.11. The case g = 13. By Mukai [43], general K3 surfaces of genus 13 can be constructed as follows. Consider the Grassmannian G(3, 7) and let E be the dual of the universal sub bundle. We take a general 2-dimensional subspace of H 0 ( 2 E) ≃ 2 (C 7 ) ∨ , which is unique up to the action of SL 7 , and let Y ⊂ G(3, 7) be its common zero locus. Then Y is a Fano 6-fold of index 3, and its stabilizer G Y in SL 7 has dimension 10. Let F be the third exterior power of the universal quotient bundle over G (3,7), and F Y = F | Y be its restriction to Y. Then F Y has rank 4, h 0 (F Y ) = 32, and the zero locus of a general section of F Y is a K3 surface of genus 13. The resulting moduli map PH 0 (F Y )/G Y F 13 is dominant and its general fibers are birationally K3 surfaces (Fourier-Mukai partner of the original K3 surface).
Proof. Let U ⊂ PH 0 (F Y ) be the open set of sections τ whose zero locus S τ is a K3 surface. Consider the incidence correspondence X n = { ([τ], p 1 , · · · , p n ) ∈ U × Y n | p i ∈ S τ }, which is irreducible, and let π : X n → Y n be the projection. The π-fiber over p = (p 1 , · · · , p n ) ∈ π(X n ) is an open dense subset of the linear subspace PV p ⊂ PH 0 (F Y ) of sections vanishing at p 1 , · · · , p n . When n = 7, V p has dimension ≥ h 0 (F Y ) − 4 · 7 = 4, so X 7 → π(X 7 ) is birationally a P m -bundle with m ≥ 3. Since the stabilizer in G Y of general p ∈ π(X 7 ) is finite, X 7 /G Y is uniruled. Since the fibers of the moduli map X 7 /G Y F 13,7 are K3 surfaces, F 13,7 is also uniruled.
The proof shows that the fibration of Fourier-Mukai partners over F 13,7 (dual K3 fibration) is also uniruled. Note that F 13,8 is the only missing case. 5.2.12. The case g = 14. Farkas-Verra [18] proved that F 14,1 is rational using special cubic 4-folds.
5.2.13. The case g = 16. By Mukai [44], general K3 surfaces of genus 16 can be constructed as follows. Let T be the GL 4 -equivariant compactification of the space of twisted cubics in P 3 constructed by Ellingsrud-Piene-Stromme [15]. The variety T is smooth of dimension 12 and acted on by GL 4 . There exist vector bundles E, F on T of rank 3, 2 respectively and equivariant linear maps (S 2 C 4 ) ∨ → H 0 (E) and (S 2,1 C 4 ) ∨ → H 0 (F ).
Here S 2 C 4 is the symmetric square of C 4 and S 2,1 C 4 is the kernel of C 4 ⊗ S 2 C 4 → S 3 C 4 . We write V = (S 2 C 4 ) ∨ . If M ⊂ V is a general 2dimensional subspace, its common zero locus Y M has dimension 6. Let Syz M ⊂ S 2,1 C 4 be the kernel of C 4 ⊗ M ⊥ → S 3 C 4 , which has dimension 12. Then a general 2-dimensional subspace N of  Proof. As in the case g = 12, we consider the incidence correspondence in two steps: Z n = { (M, p 1 , · · · , p n ) ∈ G(2, V) • × T n | p i ∈ Y M }, X n = { (M, N, p 1 , · · · , p n ) ∈ P • × T n | p i ∈ S M,N }.
As in the cases g = 7, 13, the proof shows that the dual K3 fibration over F 16,4 is also uniruled. 5.2.14. The case g = 18. By Mukai [40], general K3 surfaces of genus 18 can be constructed as follows. Fix a smooth quadric Q in P 8 and let Y = OG (3,9) be the orthogonal Grassmannian parametrizing 2-planes contained in Q. We have a rank 2 homogeneous vector bundle F over Y such that V = H 0 (Y, F ) is the 16-dimensional spin representation of Spin (9). If N is a general 5-dimensional subspace of V, its common zero locus S N is a K3 surface of genus 18. The moduli map G(5, V)/SO (9) F 18 is birational.