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The moduli space of cubic fourfolds

We describe the GIT compactification of the moduli space of cubic fourfolds, with a special emphasis on the role played by singularities. Our main result is that a cubic fourfold with only isolated simple (A-D-E) singularities is GIT stable. Conversely, with some minor exceptions, the stability for cubic fourfolds is characterized by this condition.


Introduction
One of the classical results of algebraic geometry is that the moduli space of elliptic curves can be constructed both as a GIT quotient and as the quotient of the upper half plane by the modular group. Several similar examples, where a dual construction for the moduli space exists, were known classically (e.g. the low degree K3 surfaces), and a few more were discovered recently (e.g. the moduli spaces of cubic surfaces and cubic threefolds -see [3], [4] and [18]). The purpose of this and of a subsequent paper is to discuss a new example in this vein, namely, we analyze the moduli space of cubic fourfolds.
The moduli space of cubic fourfolds came to attention recently as a key ingredient in the Allcock-Carlson-Toledo [4] and Looijenga-Swierstra [18] construction of a uniformization of the moduli space of cubic threefolds by a complex ball. One important reason for this is that the period map for cubic fourfolds behaves quite nicely, very similarly to the period map for K3 surfaces. There are a series of results in this direction (e.g. Beauville-Donagi [9]) culminating with the proof of the global Torelli theorem by Voisin [28] and the work of Hassett [14] on rationality. However, in contrast to the K3 case, surjectivity type results for cubic fourfolds (i.e. the characterization of the image of the period map) are not known.
Following the example of low degree K3 surfaces (Shah [25,26]) and cubic threefolds (see [1], [31], [4] and [18]), we attack the problem mentioned above by means of geometric invariant theory. Namely, here we discuss the degenerations of cubic fourfolds from a GIT point of view. In subsequent work, these results are combined with a monodromy analysis to get a good understanding of the image of the period map for cubic fourfolds. The main idea behind this approach is that, quite generally, GIT provides a quick way of compactifying a moduli space. The resulting compactification can then be used as a starting point for more refined constructions.
The main problem, however, is that typically the GIT compactification is quite badly behaved. For example, it is quite possible that nice objects for the moduli problem under consideration are left out of the GIT compactification. Our first main result is that this type of phenomena does not occur for cubic fourfolds. Specifically, the following theorem essentially identifies the stable cubic fourfolds as those having simple singularities.

Theorem 1.1. A cubic fourfold Y is not stable if and only if one of the following conditions holds:
i) Y is singular along a curve C spanning a linear subspace of dimension at most 3 of P 5 ; ii) Y contains a singularity that deforms to a singularity of class E r (for r = 6,7,8).
In particular, if Y is a cubic fourfold with isolated singularities, then Y is stable if and only if Y has at worst simple singularities.
In particular, Theorem 1.1 allows us to speak about the moduli space of cubic fourfolds M with at worst simple singularities. This is extremely relevant in the context of analyzing the period map for cubic fourfolds. We recall that the simple singularities in even dimensions are characterized by the fact that they give finite monodromy. Therefore, the space M is the natural space where the period map (defined a priori only for smooth cubic fourfolds) would extend. The fact that the simple singularities give stable points plays a key role in the case of K3 surfaces (see Shah [25]). It is also implicitly used in the case of cubic threefolds by Allcock-Carlson-Toledo [4] and Looijenga-Swierstra [18].
Our second main result is the following description of the GIT compactification of the moduli space of cubic fourfolds: Theorem 1.2. The moduli space M of cubic fourfolds having at worst simple singularities is compactified by the GIT quotient M by adding six irreducible boundary components, that we label α, . . . , φ. A semistable cubic fourfold X with minimal orbit corresponding to a generic point in a boundary component has the following geometric property: The description of the GIT compactification M might seem complicated, but we note that there is quite a bit of structure. To start, we note that, as discussed in section 8, the GIT computation for cubic fourfolds is closely related to that for cubic threefolds (Allcock [1] and Yokoyama [31]) and that for plane sextics (Shah [25]). It follows that one can essentially reconstruct the cubic fourfold case from these two lower dimensional cases. At a deeper level, in all three cases mentioned here, the structure of the moduli space is dictated by the Hodge theoretical properties of the varieties under consideration. We only lightly touch on this in section §8. 3. Nonetheless, it is quite apparent from our computations that the relationship between the GIT construction and Hodge theoretical construction of the moduli space of cubic fourfolds is very similar to that for low degree K3 surfaces (see [25,26] and [17, §8]).
A few words about the organization of the paper. A standard GIT analysis consists of three steps. The first one is a purely combinatorial one, and consists of identifying certain maximal subsets of monomials. We discuss this step for cubic fourfolds in section 2. The next step (corresponding to sections 3 and 4 in our text) attaches some geometric meaning to the combinatorial results obtained in the previous step. The results in this step typically describe the stability of hypersurfaces in terms of a "bad flag" (see Theorem 3.2 and the discussion from [21, §4.2]). Unfortunately, this geometric interpretation is rather coarse, so one needs to refine these results. Typically, by using some classification of singularities, one can interpret the existence of bad flags in terms of singularities. In our situation, we divide the analysis in two cases: isolated (section 5) or non-isolated (section 6) singularities. The main results here are Theorems 5.6 and 6.1. We note that the most delicate aspect is the identification of the boundary strata ǫ and φ corresponding to stable cubic fourfolds with non-isolated singularities. Finally, in sections 7 and 8, we put everything together and conclude the proofs of the main results and make some further comments on the structure of the GIT compactification M.
After the completion of this work, we've learned that both Allcock [2] and Yokoyama [30] (independently) have done a partial analysis of the stability for cubic fourfolds. Our results, however, are more detailed and amenable to the study of the period map for cubic fourfolds.
1.1. Acknowledgments. While writing this paper, I've benefited from discussions with Igor Dolgachev, Bob Friedman, and Rob Lazarsfeld. I am grateful for their helpful comments and support. I would also like to thank Eduard Looijenga for informing me about his computation of the Baily-Borel compactification and the work of Yokoyama.
1.2. Notations and Conventions. We fix the following notations: -M 0 : the moduli space of smooth cubic fourfolds; -M : the moduli space of cubic fourfolds with simple singularities; -M s : the moduli space of stable cubic fourfolds; -M : the GIT compactification of M 0 . Let W (= H 0 (P 5 , O P 5 (1))) be a six dimensional vector space. By definition, where the quotient is taken in the sense of GIT [21]. The GIT terminology is that of Mumford [21]. For us, unstable means not semistable, non-stable is a shorthand for not properly stable, and strictly semistable means semistable, but not properly stable. As is customary, the one parameter subgroups (1-PS) λ of SL(6) that are used in the application of the numerical criterion are assumed diagonal t ∈ C * λ − → diag(t a0 , . . . , t a5 ) ∈ SL(6) with the weights satisfying a 0 ≥ · · · ≥ a 5 and a 0 + · · · + a 5 = 0. We call such a 1-PS normalized and denote it by its weights (a 0 , . . . , a 5 ). Given a monomial x i0 0 . . . x i5 5 , its weight with respect to a normalized 1-PS λ of weights (a 0 , . . . , a 5 ) is i 0 · a 0 + . . . i 5 · a 5 .
We denote by M ≤0 (λ) (and M <0 (λ)) the set of monomials of degree 3 which have non-positive (resp. negative) weight with respect to λ. Similarly, M 0 (λ) is set of monomials of weight 0 (i.e. invariant with respect to λ).
We make the convention that all the boundary strata (α, β, . . . ) that occur in our classification are closed and irreducible. In particular, it makes sense to discuss about a generic point in a boundary stratum. By an adjacency of strata (represented by an arrow in figures 1 and 2), we understand an inclusion of strata. The various symbols used in figure 1 and 2 represent the dimensions: •, •, 2, and ⋄ mean 1-, 2-, 3-, and 4-dimensional boundary components respectively.
We are concerned with the following analytic types of isolated hypersurface singularities: A n (n ≥ 1), D n (n ≥ 4), E r (r = 6,7,8), and E r (r = 6,7,8). The singularities of type A n , D n , and E r are called simple singularities. The terminology and notations are those of Arnold [7], with the exception that we use E 6 , E 7 , E 8 instead of P 8 , X 9 , and J 10 respectively. We also consider two types of non-isolated hypersurface singularities: A ∞ (double line) and D ∞ (pinch point) given locally in (C 5 , 0) by the equations (x 2 2 + · · · + x 2 5 = 0) and (x 1 x 2 2 + · · · + x 2 5 = 0) respectively. For the reader familiar with these notations for surface singularities, we mention that we refer to the fourfold singularities obtained by suspending the homonymous types of surface singularities.
By the corank of the hypersurface singularity given by f (x 1 , . . . , x n ), we understand the number of variables n minus the rank of the Hessian of f (see [7,Ch. 11]). The corank is a stable invariant of the singularity. We note that a hypersurface singularity of corank 1 is of type A n (for some n ≥ 2) if isolated, or A ∞ otherwise (see [7,Ch. 11,16]). Similarly, a singularity of corank 2 and third jet x 2 1 x 2 is of type D n (for some n ≥ 5) or D ∞ .

Preliminary Study of Stability
The main tool of analyzing the stability for cubic fourfolds is the Hilbert-Mumford numerical criterion [21,Thm. 2.1]: A cubic form f is stable (resp. semistable) iff µ(f, λ) > 0 (resp. ≥ 0) for all λ one parameter subgroups of SL (6), where µ(f, λ) is the numerical function of Mumford. As is customary, we fix coordinates on P 5 and assume that all 1-PS used are normalized as in §1.2 (equivalently, fix a maximal torus T in SL (6), and consider only 1-PS of T ). In order to understand the stable cubic fourfolds, it suffices to find the cubic forms f for which there exists a normalized 1-PS λ such that µ(f, λ) ≤ 0 (any other non-stable cubic is projectively equivalent to such an f ). By definition (see [21, pg. 81]), µ(f, λ) is the highest weight with respect to λ of a monomial occurring with non-zero coefficient in f . Thus, the cubic defined by f is unstable if all the monomials in f belong to M ≤0 (λ) for some λ. It is then clear that, for the analysis of stability, it is enough to identify the maximal possible subsets M ≤0 (λ) and to interpret geometrically what it means for f to be a linear combination of monomials from M ≤0 (λ). In our situation, we identify the maximal M ≤0 (λ) by applying a simple computer program (see remark 2.1). The results are given in table 1 below. We note that similar computations occur in literature (see esp. [1] and [31]).
Remark 2.1. Let us briefly indicate the procedure of finding the maximal subsets M ≤0 (λ). We ask the equivalent question of finding the maximal subsets M of degree 3 monomials in 5 variables for which there exists a 1-PS λ such that M ⊆ Case Weights of λ Maximal monomials Invariant S1 (2, 2, −1, Proof. An equation of type S6 has the general form: and it is characterized by the fact that (1 : 0 : · · · : 0) is a singular point of corank 3 for the fourfold defined by g 1 (see also 3.7). To prove the lemma, it suffices to find a singular point of corank at least 3 for an equation of type S7 or S8. In the case S7, we write the corresponding equation as: The point (1 : 0 : · · · : 0) is a singularity of corank at least 3 for g 2 . The reduction of g 2 to the form S6 is then obvious. The case S8 is similar.
For a cubic form f such that µ(f, λ) ≤ 0, the limit lim t→0 f t = f 0 exists and it is invariant with respect to λ. For the six cases S1-S6, we denote the corresponding invariant parts by α, . . . , δ. A general equation of type α, . . . , δ has the form: where the dual λ * means opposite weights. Thus, the cases S1-S6 come in pairs. The case S3 is self-dual. The cases S6-S8 can be reduced (and are in fact equivalent) to the single case S6 (cf. lemma 2.2) due to the fact that an equation of type δ is stabilized by a 2-dimensional torus.
Similarly to case of stable cubics, in order to understand the semistable cubics one has to find the maximal subsets of type M <0 (λ) as λ varies over all normalized 1-PS. We solve this problem by applying a simple modification of the algorithm described in remark 2.1. The results are given in table 2 below. Each of the maximal subsets M <0 (λ) is included in some maximal subset M ≤0 (λ ′ ) (see the last column of table 2). Again, some of these cases can be removed by using projective equivalences.  (11,11,11, −1, −7, −25) We conclude the combinatorial aspect of the stability analysis by summarizing the results of the section in the following proposition: Proposition 2.6. A cubic fourfold is not properly stable (resp. unstable) if and only if it is of type Sk (resp. Uk) for some k = 1, . . . 6. Furthermore, a strictly semistable cubic fourfold with minimal orbit is of type α-δ.

Geometric interpretation of stability
In this section, we start the geometric analysis of the stability condition for cubic fourfolds. Here, we interpret the results of the tables 1 and 2 in terms of the existence of "bad flags" (Mumford [21, pg. 80]).
To explain the occurrence of flags in the geometric analysis, let us recall that a cubic fourfold Y fails to be stable if there exists a 1-PS λ such that µ(f, λ) ≤ 0, where f is the corresponding cubic form. The natural action of the 1-PS λ on the vector space W = H 0 (P 5 , O P 5 (1)) determines a weight decomposition W = ⊕ k W k . We define a flag (F n ) n of subspaces of W by F n := ⊕ k≤n W k . The condition that Y is not stable with respect to λ imposes some specific geometric properties of Y with respect to the associated projective flag P(F n ) (e.g. typically Y is singular along the linear subspace corresponding to the first term of the flag). Conversely, certain geometric properties of Y determine a special flag, which in turn leads to existence of a destabilizing 1-PS. We note that the complexity of the analysis of the cases S1-S6 depends on the length of the flag associated to λ (i.e. the number of distinct weights of λ). Additionally, a non-stable cubic fourfold Y degenerates to a cubic fourfold of type α-δ (see table 1).
Proof. As a consequence of the numerical criterion, we only need to find geometric characterizations for the cubic fourfolds of type S1-S6 (see Prop. 2.6). This is done in lemmas 3.3-3.8 below. Lemma 3.3 (Case S1). Let Y be a cubic 4-fold of type S1. Then Y contains a double line. Conversely, if Y contains a double line then Y is of type S1.
Proof. The equation of Y is of type S1, so it can be written as: The fourfold Y contains the line L : (x 2 = x 3 = x 4 = x 5 = 0) with multiplicity 2. The converse is also clear. Note that the only monomials missing from the above equation are the 16 monomials which are not in the square of the ideal x 2 , . . . , x 5 .
The proof of the following lemmas is similar, we omit the details.
Additionally, Y is singular along a curve which is the complete intersection of two quadrics in P 3 . This property is an equivalent geometric characterization of a fourfold of type S2.
Lemma 3.5 (Case S4). Let Y be a cubic 4-fold of type S4. Then Y has a singularity of corank (at least) 2 and it contains the null plane associated to this singularity. Conversely, if Y contains a singularity of corank 2 and it also contains the null plane of this singularity, then Y is of type S4.  In all the cases except S3, the destabilizing 1-PS λ has at most 3 different weights. Thus, the flag associated to λ is rather simple. As a consequence, the analysis of the previous lemmas is straightforward. In contrast, in the case S3, the associated 1-PS has 5 distinct weights, producing a flag p ∈ L ⊂ Π ⊂ H ⊂ P 5 , where p is point, L a line, Π a 3-plane, and H a hyperplane. It follows that the geometry of a cubic fourfold of type S3 is quite complicated, reflecting the complexity of the flag.
. Then Y has a singularity p ∈ Y of corank 2 with the property that there exists a hyperplane H such that: i) the intersection of H with the projective tangent cone to p (a rank 3 quadric in P 5 ) is a 3-plane Π (with multiplicity 2); ii) the restriction of Y to H contains a double line L passing through p; iii) the restriction of Y to Π consists of 3 planes meeting in the line L.
Conversely, if a cubic fourfold Y has these properties then Y is of type S3. Proof. A fourfold Y of type S3 has the defining equation: The quadric Q has rank at most 3. It follows that p = (1 : 0 : · · · : 0) ∈ P 5 is a singular point of Y of corank 2 or more. We assume that Y is not of type S6. In particular, the corank at p is exactly 2 (i.e. a = 0, and l(x 2 , x 3 , x 4 , x 5 ), x 4 and x 5 are linearly independent). The null plane of the singularity at p is given by The 1-PS λ associated to Y of type S3 singles out the line L : (x 2 = x 3 = x 4 = x 5 = 0), the 3-plane Π : (x 4 = x 5 = 0), and the hyperplane H : (x 5 = 0). Together with p and P defined above, we obtain a full flag p ∈ L ⊂ P ⊂ Π ⊂ H. The following properties of the flag are immediate: i) H meets the projective tangent cone to p (given by the quadric Q in P 5 ) in the 3-plane Π; ii) the line L is a double line for the cubic 3-fold X = Y ∩ H ⊂ H ∼ = P 4 (note that all monomials of g are in the ideal x 2 , x 3 , x 4 2 + x 5 ); iii) the restriction of Y to Π ∼ = P 3 consists of three planes meeting in the line L (note that the restriction is given by . We obtain one direction of the lemma. Conversely, we assume that Y has the geometric properties stated in the lemma. These properties recover the flag p ∈ L ⊂ Π ⊂ H, which in turn allows us to find coordinates such that Y has an equation of type S3. Specifically, let p = (1 : 0 : · · · : 0) ∈ Y be the corresponding singular point. Since p has corank 2, we can write the equation of Y as: . The null plane of the singularity at p is given by P : (x 3 = x 4 = x 5 = 0). Let H be a hyperplane as in the lemma. The assumption i) on Y forces H to contain the null plane P . Furthermore, we can view H as being spanned by P and a tangent line to the conic (x 2 4 + x 3 x 5 = 0) in the plane P ′ : (x 0 = x 1 = x 2 = 0). In particular, we get a partial flag p ∈ P ⊂ Π ⊂ H, where Π is the reduced intersection of H with the projective tangent cone at p. By a change of variables involving only x 3 , x 4 , and x 5 , and fixing the conic x 2 4 + x 3 x 5 we can assume that Π is given by Π : (x 4 = x 5 = 0) and H is given by H : (x 5 = 0). The assumption ii) on Y , implies in particular that the line L through p lies in the null plane P . Thus, we obtain a full flag p ∈ L ⊂ P ⊂ Π ⊂ H. By a change of variables involving only x 1 and x 2 we can further assume that L is given by L : (x 2 = x 3 = x 4 = x 5 = 0). With these normalizations, it is easily seen that the condition iii) is equivalent to vanishing of the coefficients of the monomials: Using again the condition ii), we obtain that the coefficient c in the equation of F vanishes. In conclusion, Y is of type S3. Proof. This is the same as lemmas 2.2 and 3.7.
Proof. If Y is unstable, it is of one of the types U1-U6. In all these cases, except U4, there exists a singularity at p = (1 : 0 : · · · : 0) of corank at least 3, or of corank 2 and third jet x 3 1 . It follows that the singularity at p is worse than D ∞ . In the case U4, Y is singular along a conic with singularities of A 2 transversal type (i.e. the singularities of a generic hyperplane section of Y are of type A 2 ). Thus, the singularities of Y are worse than D ∞ .

The minimal orbits and their normal forms
The geometric invariant theory compactifies the geometric quotient M s by adding boundary components that parametrize strictly semistable cubic fourfolds with minimal orbits. By proposition 2.6, we know that such a cubic fourfold is of type α-δ. It follows that there are four boundary components for M \ M s , which, by abuse of notation, we label also by α-δ. In this section, we discuss which of the fourfolds of type α-δ are in fact semistable, and the structure of the resulting boundary components. As noted in §3.2, a direct approach based on the numerical criterion is cumbersome. We use instead the following criterion of Luna [19,Cor. 1]: Let X be an affine G-variety, and x ∈ X a point stabilized by a reductive subgroup H.
We apply Luna's criterion for the affine space X = Sym 3 (W ) and the connected component H of the stabilizer of a general cubic fourfold of type α-δ. The advantage of this approach is that it allows us to check that an orbit of type α-δ is closed by using smaller subgroups of G = SL(W ) ∼ = SL (6). In other words, in order to understand the boundary it suffices to study the simpler action of N G (H)/H on X H instead of the full action of G on X (N.B. H acts trivially on X H ). We note also that the natural morphism X H /N G (H) → X/G is finite (Luna [19,Main Thm.]) and that Luna's criterion can be applied by using the centralizer C G (H) instead of the normalizer N G (H) (see [23, pg. 221-222]). As an immediate application of Luna's results, we obtain the dimensions of the boundary strata.
Lemma 4.1. A generic cubic fourfold of type α-δ gives a closed orbit. Therefore, each of the types α-δ gives an irreducible boundary component for the GIT compactification M s ⊂ M. The dimensions of these boundary strata are 1 for α and δ, 2 for γ, and 3 for β.
Before starting the detailed analysis of the strata α-δ, we observe that the types β and γ have as a common specialization the curve: The stabilizer of an equation of type τ contains the 2-dimensional torus generated by λ : (2, 1, 0, −1, −2, 0) and λ ′ : (4, 1, 1, −2, −2, −2). The equation (4.1) further degenerates (for a = 0) to: which is stabilized by a 4-dimensional torus. The orbit ζ is also a specialization of the cases α and δ. The resulting incidence diagram is given in figure 1.
y y r r r r r r r r r r r r r τ ζ The determinantal cubic is singular along the Veronese surface and is stabilized by a subgroup of SL(6) isomorphic to SL(3).
Our first result regarding the minimal orbits establishes the semi-stablity of the degenerate cases τ and ζ.
We now do the case-by-case analysis of the minimal orbits of type α-δ. The common feature of all these cases is that the analysis reduces to some well-known lower dimensional GIT problem. For example, the case α reduces to the analysis of the stability for pencils of quadrics (see [29] and [8]).
The fourfold Y is singular along E and along the line L : (x 2 = · · · = x 5 = 0). The singularities along E are of type A ∞ . Similarly, those along L are of type A ∞ at all but 4 points (the zeros of ∆), where they become of type D ∞ . ii) If ∆(x 1 , x 2 ) has a double root, then Y degenerates to ζ. iii) If ∆(x 1 , x 2 ) has a triple root (or vanishes), then Y is unstable.
Proof. The connected component of the stabilizer of an equation of type α is the 1-PS H of weights (2, 2, −1, −1, −1, −1). It follows that C G (H) acts on an equation of type α as SL(2) × SL(4): SL(2) acts on the variables x 0 and x 1 , and SL(4) acts on x 2 , . . . , x 5 . More intrinsically, the action of C G (H)/H (N.B. H is abelian) on X H is equivalent to the natural action of SL(2) × SL(4) on V ⊗ Sym 2 (U ), where V and U are the standard representations of SL(2) and SL(4) respectively. Therefore, the GIT analysis for a fourfold of type α is equivalent to the GIT analysis for the pencil of quadric surfaces in P 3 given by q 1 and q 2 (see [29, §2]). According to Wall [29, §4(a)], the pencil is semi-stable if and only if the multiplicity of the roots of ∆(x 1 , x 2 ) is at most 2. If the roots of ∆ are distinct, the corresponding orbit is closed and the quadrics q 1 and q 2 can be simultaneously diagonalized, giving the equation (4.4) (see [8,Prop. 2]). The invariant of the pencil is the base locus, the elliptic curve E. If ∆ has a double root, then the corresponding orbit contains in its closure the orbit ζ.
The case δ is quite similar, we omit the details.
Lemma 4.5. Let Y be a fourfold of type δ (see (2.4)). One of the following holds: x 2 , x 3 ) has a node, then Y degenerates to ζ; iv) if none of the items i-iii are true, then Y is semistable with closed orbit.
After a linear change of coordinates we can assume Y is given by: is a smooth plane cubic. The fourfold Y has 3 singularities of type E 6 (of the same modulus and non-colinear).
As the dimension of stratum increases, the analysis gets more involved.
Lemma 4.6. Let Y be a fourfold of type γ given by (2.3). Assume that the rank of the quadric q is 3 (otherwise is of type S6 and can be treated as such). Let . Then one of the following holds: i) The conics q and q 2 meet in 4 distinct points. Then Y gives a minimal orbit and its equation can be taken as . Therefore, the invariant part is the determinant q 2 . We obtain two conics q and q 2 in P 2 . By acting with SL(3) we can bring the two conics to the normal form (4.6). The remaining part follows by an analysis of the possible degenerations.
The last remaining case is the 3-dimensional boundary stratum β.
Lemma 4.7. Let Y be a fourfold of type β given by (2.2). The orbit of Y is not closed iff any of the following conditions are satisfied: a, b, l 1 , or f vanish, l 1 is a factor of f , or l 2 2 is a factor of f (including the case f has a double root and l 2 vanishes). If the orbit of Y is not closed, then either Y is unstable or Y degenerates to a fourfold of type τ (including ζ). If Y is semistable with closed orbit, then: i) Generically, Y has two isolated singularities of type E 8 (of same modulus). ii) If Y doesn't have isolated singularities, then Y is singular along a rational normal curve of degree 4 (the case labeled σ (6.3) and discussed in §6.2).
Proof. Let H be the 1-PS (2, 1, 0, 0, −1, −2) that stabilizes Y . The center of C G (H) is a 4-torus acting on an equation of type α by scaling of the variables. The factor group C G (H)/Z(C G (H)) is isomorphic to PGL(2) acting on l 1 , l 2 , and f by linear changes of the variables x 2 and x 3 . This makes the structure of X H /C G (H) (and consequently that of strata β) quite transparent. In particular, in order to understand the closed orbits, we can apply a slight modification of the computer program used in §2. As a result, we obtain the degeneracy conditions of the lemma (N.B. most of these conditions can be seen directly). By inspecting (2.2), it is easy to see that there are two singularities of corank 2 at (1 : 0 : · · · : 0) and (0 : · · · : 0 : 1). In the generic case, they are of type E 8 (see also section 5, esp. the proof of 5.5). Due to the obvious symmetry, the singularities have the same modulus. If Y does not have isolated singularities, then Y is singular along a curve C. We distinguish two cases, either C is contained in a linear 3-dimensional subspace of P 5 or not. The former case gives a non-closed orbit which degenerates to the case τ (see also §6.1). Thus, by theorem 6.1, we can assume that C is either a rational normal curve of degree 4 or an elliptic normal curve of degree 6. Since Y and (consequently) C are stabilized by a subgroup of SL(6) isomorphic to C * , the elliptic case is excluded. We conclude that C is a rational normal curve of degree 4.

Stability of cubic fourfolds with isolated singularities
In this and following section, we refine the results of section 3, by interpreting the failure of stability for cubic fourfolds in terms of the existence of bad singularities (rather than bad flags). The main tool, as noticed by Allcock [1, §2], is the following result (see [7, pg. 209]) that identifies the analytic type of certain classes of singularities: If an analytic function f (x 1 , . . . , x n ) is semiquasihomogeneous with respect to the weights corresponding to an A n , D n , E r or E r (for r = 6, 7, 8) singularity, then f has a singularity of that type at the origin.
We recall that a convergent power series f (x 1 , . . . , x n ) is semiquasihomogeneous with respect to a choice of weights if the leading term f 0 (x 1 , . . . , x n ) (w.r.t. the weighting) defines an isolated singularity at the origin. The weights that make the A-D-E singularities quasihomogeneous are well known (e.g. [1, §2] It follows that the cubic fourfolds with very mild singularities are stable.

Corollary 5.3. A cubic fourfold having at worst simple singularities is stable.
Proof. Let Y 0 be such a fourfold. Assume that Y 0 is not properly stable. According to proposition 2.6, the fourfold Y 0 is of one of the types S1-S6. If the type is S1, S2, or S5, then Y 0 is singular along a curve (cf. theorem 3.2), thus a contradiction. It remains to consider the cases Sk for k=3, 4, or 6. Choose coordinates such that the equation g of Y 0 is as given by table 1. It follows that p = (1 : 0 . . . , 0) ∈ Y 0 is a singular point of Y 0 . By considering a general deformation of Y 0 that preserves the Sk type (w.r.t. the given choice of coordinates), we see that the singularity at p deforms to a singularity of type E r (cf. 5.2 above). This is a contradiction to the well known fact that no A-D-E singularity deforms to an E r singularity. We conclude that a cubic fourfold simple (isolated) singularity is stable.
The following lemmas establish a converse to the previous corollary: Lemma 5.4. Let Y be a cubic fourfold having a singularity that deforms to E 6 . Then Y is of type S6. In particular, Y is not stable.
Proof. A singularity that deforms to E 6 has corank at least 3 (by the semi-continuity of the corank). Therefore, Y is of type S6 (cf. lemma 3.7).
Lemma 5.5. Assume that the cubic fourfold Y has an isolated singularity of corank 2 that deforms to E 8 or E 7 , then Y is of type S3 or S4. In particular, Y is not stable.
Proof. An equivalent formulation of the lemma is: if p ∈ Y is a singularity of corank 2, then either p is of type D n or E r , or Y is of type S3 or S4. This follows from a careful analysis of the position of null plane of the singularity at p and a systematic application of theorem 5.1, as sketched below. Without loss of generality, we assume that the singularity p ∈ Y of corank 2 is at (1 : 0 · · · : 0) ∈ P 5 . For an appropriate choice of coordinates we can write the equation g of Y as: g(x 0 , . . . , . the rank of the tangent cone is 3). We project Y onto a hyperplane, say (x 0 = 0), not passing through the singularity. In this hyperplane we take the coordinates (x 1 : · · · : x 5 ), and denote by Q the projectivized tangent cone (the quadric given by x 2 4 − x 3 x 5 ), and by X the cubic threefold given by F (x 1 , . . . , x 5 ). The null plane P of the singularity projects to the lineP : (x 3 = x 4 = x 5 = 0), the singular locus of the quadric Q. We have the following possibilities for the position of the lineP relative to X: (a)P is transversal to X; (b)P meets X with multiplicity 2 in a point; (c)P meets X with multiplicity 3 in a point; (d)P is contained in X.
In the first two cases, Y has the third jet either x 3 1 + x 3 2 or x 1 x 2 2 . Thus, the singularity at p is of type D n (see [7, pg. 190]). For instance, in the case (a) the equation (in affine coordinates) of the singularity at p can be taken to be: x 2 4 − x 3 x 5 + x 3 1 + x 3 2 + (higher order terms). Thus, p is of type D 4 (cf. theorem 5.1).
The case (d) is equivalent to saying that the null plane P of the singularity at p is contained in the fourfold Y . By lemma 3.5, Y is of type S4 and we are done.
The only remaining case is (c). In this situation, either Y has a singularity of type E r for r = 6, 7, 8 at p, or it satisfies the degeneracy conditions of lemma 3.8. First, one verifies that unless both conditions i) and ii) of lemma 3.8 (they impose a single additional condition to (c)) are satisfied then Y has a singularity of type E 6 at p. Given that i) and ii) are satisfied, the condition iii) differentiates between the case when singularity at p is E 7 or E 8 and the case when the singularity is E 8 (or worse). Specifically, the projection of the 3-plane Π of the lemma 3.8 is a 2-planeΠ which cuts on X a singular plane cubic curve C ⊂Π. If C is nodal or cuspidal the singularity at p is of type E 7 or E 8 respectively (again by applying theorem 5.1). Due to some geometric restrictions on C imposed by i) and ii), the only possibility for C to be reducible is to consist of 3 lines meeting in one point. This situation is equivalent to the condition iii) of the lemma 3.8, and generically (if the three lines are distinct) gives an E 8 singularity. We obtain that, in the case (c), either the singularity at p is of type E r or Y is of type S4, concluding the proof of the lemma.
We conclude: Theorem 5.6. Let Y be a cubic fourfold with isolated singularities. Then Y is stable if and only if it has at worst simple singularities.
Proof. One direction is given by 5.3 above. Conversely, any non-simple isolated singularity deforms to a singularity of type E r singularities. Thus, the result follows from lemmas 5.4 and 5.5.
Remark 5.7. We make the following remarks on the extent to which the stability can be characterized in terms of singularities.
i) An important reason why a result such as theorem 5.6 is possible is that much of the information needed to decide that the singularity is simple is given by corank and third jet. On the other hand, in the case of cubics, the same invariants are the first that come up in the GIT analysis. ii) It is not hard to extend the results of this section to include also the singularities E r . We can show that a cubic fourfold with at worst E r is semi-stable. Additionally, if Y contains a singularity of type E r , then Y degenerates to a fourfold of type β, γ, or δ for r = 8, 7, or 6 respectively. iii) On the negative side, it does not seem to us that it is possible to extend the results beyond the E r singularities. In particular, it is unlikely that reasonable analogous statements to those of Allcock [

Cubic fourfolds with non-isolated singularities
In order to complete the geometric analysis of stability for cubic fourfolds, we have to understand the cubic fourfolds with non-isolated singularities. More precisely, we are interested in finding the cubic fourfolds with non-isolated singularities that are stable, and a parameterization for them. This is answered by the following theorem that summarizes the results of the section. Proof. Let C be an irreducible curve included in the singular locus of Y . If C is contained in a 3-dimensional linear subspace of P 5 , we are essentially done by theorem 3.2. The detailed discussion is done in §6.1. The remaining cases, C is contained in a hyperplane or C is linearly non-degenerate, are discussed in §6.2 and §6.3 respectively. These non-degenerate cases produce the new strata ǫ and φ.
Remark 6.2. We note that there are several results in literature that partially overlap with the results of theorem 6.1 (see especially Aluffi [5] and O'Grady [22]).
The general equations for the cubic fourfolds parametrized by the strata ǫ and φ of the theorem are: for a, b, c ∈ C. Inside the surface σ, there are two curves that parameterize degenerate geometric situations. Namely, the curve τ (see (4.1)) that was discussed in section 4, and the curve χ: for a, b ∈ C. The curves τ and χ intersect in the point ω (see (4.3) and remark 4.2), which is in some sense the most degenerate point of the GIT quotient. The resulting adjacency diagram is given in figure 2. For further details on the strata ǫ and φ, we refer the reader to §6.2 and §6.3 respectively.
• x x r r r r r r r r r r r r r r Figure 3. The stabilizers of general points in the boundary strata Remark 6.3. The boundary strata for a strict GIT analysis are given in figure 1. Figure 2 is obtained from figure 1 by adding information about the possible singularities of a cubic fourfold. It is interesting to note that much of the extra structure of figure 2 can be also obtained by taking into account the possible stabilizers of cubic fourfolds (compare figures 2 and 3). The stratification given by stabilizers plays a key role in the analysis of the geometry of M from a GIT point of view (see the work of Kirwan [16,15]).

Strictly semistable cubic fourfolds with non-isolated singularities.
According to theorem 3.2 a cubic fourfold containing a line, a conic or a rational elliptic curve of degree 4 is not stable. Here, we prove that these are all the possibilities for a cubic fourfold to be singular along a curve contained in a 3-dimensional linear subspace of P 5 . Proposition 6.4. Let Y be a cubic fourfold with dim Sing(Y ) = 1. Assume that an irreducible 1-dimensional component C of the singular locus is contained in a 3-dimensional subspace of P 5 . Then, C is either a line, a conic, or a component of the complete intersection of two quadrics in P 3 . In particular, Y is non-stable of type S1, S5, or S2 respectively. A general cubic Y singular along a line, a conic, or an elliptic normal curve of degree 4 is strictly semistable with the closure of its orbit containing a closed orbit of type α, γ, or α respectively.
Proof. Let C ⊂ Sing(Y ) be a curve as in proposition, and denote by Π := Sec(C) its secant variety. It is well known that a cubic contains the secant variety of its singular locus. Thus, Π ⊂ Y . Since the secant variety of a curve has always the expected dimension, it also follows that Π is a linear subspace of P 5 , coinciding with the linear span of C. Now the claim follows easily. For example, say dim Π = 3. Under appropriate choice of coordinates, we can assume Π is given by (x 4 = x 5 = 0). It follows that the defining equation of Y can be taken to be: g(x 0 , . . . , x 5 ) = x 4 Q 1 (x 0 , . . . , x 5 ) + x 5 Q 2 (x 0 , . . . , x 5 ), for some quadrics Q 1 and Q 2 . Let q 1 (x 0 , . . . , x 3 ) and q 2 (x 0 , . . . , x 3 ) be the quadrics obtained by restricting Q 1 and Q 2 to Π. If both q 1 and q 2 vanish, we get dim Sing(Y ) ≥ 2. Otherwise, it is easily seen that the intersection of Π with the singular locus Sing(Y ) is the complete intersection of the quadrics q 1 and q 2 . The conclusion follows.
The stability of cubic fourfolds with singular locus at least 2-dimensional is settled by the following proposition: Proposition 6.5. Let Y be a cubic fourfold with dim Sing(Y ) ≥ 2. Then one of the following holds: i) the singular locus contains a plane or a quadric surface; ii) the singular locus is the cone over the rational normal quartic, in which case Y is the cone over the chordal cubic threefold (unique up to projective equivalence); iii) the singular locus is the Veronese surface in P 5 , in which case Y is the secant variety of the Veronese surface (unique up to projective equivalence). Furthermore, Y is semistable if and only if it is the secant variety of the Veronese surface, in which case it gives the minimal orbit ω.
Proof. The case of dim Sing(Y ) = 3 is easily settled (Y is singular along a plane and unstable). Assume dim Sing(Y ) = 3 and let S be an irreducible surface in the singular locus. By cutting Y with a generic hyperplane we obtain a cubic threefold singular along the irreducible curve C cut on S by this hyperplane. It is known that the degree of C (and thus S) is either 1, 2 or 4 (cf. [31, Prop. 4.2]). For degree 1 or 2 the proposition follows immediately.
If S is of degree 4, we can assume that S is non-degerate. Otherwise, one obtains a contradiction. It follows that S is a surface of minimal degree in P 5 , i.e. S is: -a smooth rational quartic scroll, -the Veronese surface, -or the cone over the rational normal quartic curve in P 4 . (see [11, pg. 523]). Since Y is singular along S, we have Sec(S) ⊂ Y . The expected dimension of the secant variety of a surface is 5. For smooth surfaces, the expected dimension is the actual dimension with a single exception, the Veronese surface (see [11, pg. 179]). Thus, if S is smooth, we must have that S is the Veronese surface and Y is the secant variety of S. As discussed in §4, the fourfold Y gives the minimal orbit ω. Finally, if S is singular (i.e. the cone over the rational normal curve), it is easily seen that Y is the cone over the chordal cubic threefold. It follows that Y is unstable of type U6 (cf. lemma 3.9).
6.2. Cubic fourfolds singular along a rational normal curve of degree 4. We now consider the case of cubic fourfolds singular along a curve that is contained (and is non-degenerate) in a hyperplane in P 5 . We establish that a general fourfold of this type is stable, and that the corresponding locus in the moduli space of cubic fourfolds M is irreducible and 3-dimensional. The key to this result is that we can find a normal form for such fourfolds for which the projective invariants are clear. Proposition 6.6. Let Y be a cubic 4-fold with dim Sing(Y ) = 1. Assume C ⊆ Sing(Y ) is an irreducible curve whose linear span is 4-dimensional. Then C is a rational normal quartic in P 4 . The equation of Y can be taken of the form (6.1).
Proof. Let H be the hyperplane spanned by C and X the cubic threefold obtained by restricting Y to H. By construction, X is singular along the curve C. Since C is non-degenerate in H, it follows that X is irreducible. By the analysis of cubic threefolds with non-isolated singularities (e.g. [31, Prop. 4.2]), it follows that C is a rational normal curve.
To establish the normal form (6.1) for Y , we note that X is the secant variety of C, i.e. X is the chordal cubic 3-fold (see [1, §1]). Thus, we can choose coordinates such that the equation of Y is given by (6.5) g(x 0 , . . . , x 5 ) = F 0 (x 0 , . . . , x 4 ) + x 5 Q(x 0 , . . . , x 4 ) + x 2 5 l(x 0 , . . . , x 4 ) + bx 3 5 , where F 0 is the equation of the chordal cubic and the hyperplane H is given by (x 5 = 0). The condition that C is in the singular locus of Y is equivalent to asking that the quadric given by Q(x 0 , . . . , x 4 ) in H ∼ = P 4 contains C. There exists a 6-dimensional linear system of quadrics containing the rational normal C (i.e. dim H 0 (P 4 , I C (2)) = 6). By using linear changes of coordinates, one sees that Q is defined up to addition of elements from the Jacobian ideal of F 0 . The partials of F 0 spans a 5-dimensional subspace of H 0 (P 4 , I C (2)). Thus, there is no loss of generality in assuming that in (6.5) we have Q = aQ 0 for any quadric Q 0 containing C and not lying in the Jacobian ideal. There is in fact a canonical choice for Q 0 (up to scalling). Namely, we choose Q 0 to be the unique quadric left invariant by SL(2) acting on P 4 ∼ = Sym 4 P 1 via the action induced by the natural action on P 1 (see [13,Ex. 10.12]). Concretely, Q 0 (x 0 , . . . , x 5 ) = 4x 1 x 3 − 3x 2 2 − x 0 x 4 and the normal form (6.1) is obtained immediately.
We have proved that any cubic fourfold singular along a curve spanning a hyperplane in P 5 can be put in the normal form (6.1). There are two types of transformations preserving the equation (6.1): the scaling of the variable x 5 and the action of SL(2) on the variables x 0 , . . . , x 4 by means of the isomorphism P 4 ∼ = Sym 4 P 1 (induced by the rational normal curve of degree 4). It follows that (up to projective transformations) the cubics of type ǫ depend on 3 parameters. One of these parameters is quite geometric: it is the j-invariant of the 4 points cut on the rational normal curve (the singular locus of the cubic) by the linear form l. Expanding on this observation, we obtain: Proposition 6.7. Let Y be a cubic fourfold singular along a rational normal curve C of degree 4. Assume that defining equation of Y is of the form (6.1). Then, one of the following holds: i) The hyperplane given by l cuts C in four distinct points. In this situation Y is stable. The singularities of Y along C are of type A ∞ , except at the 4 points where l cuts C. At these points the singularities are of type D ∞ . ii) The hyperplane given by l is simply tangent to C in some point. Then, the orbit of Y contains in its closure the orbit of a fourfold of type σ. A general point of σ corresponds to a semi-stable cubic fourfold with closed orbit (otherwise, it degenerates to a fourfold of type τ ). iii) The hyperplane given by l intersects C with multiplicity at least 3, or l is identically 0. Then, the orbit of Y contains in its closure the orbit of a fourfold of type χ. For a and b not simultaneously 0 in (6.4), a fourfold of type χ is semi-stable with closed orbit. iv) If a = b = 0 and l ≡ 0 then Y is the cone over the chordal cubic threefold and it is unstable.
Proof. The stability statement follows by analyzing the possible singularities of Y , and concluding that, in the case i), Y does not satisfy any of the conditions of theorem 3.2. If the hyperplane given by l becomes tangent to C, the fourfold Y is no longer stable; it is of type S3 (with respect to the 1-PS of weights (2, 1, 0, −1, −2, 0)). It then follows that Y degenerates to a fourfold of type σ. The case σ is a particular case of β and as such one can determine precisely when the orbit is closed (see the first part of lemma 4.7). In particular, it follows that the surface σ meets the other boundary components along the curve τ . Similar arguments apply to the case χ.
Remark 6.8. Strictly speaking, one does not need to separate the case χ from the case σ. However, we choose to consider it as a separate case for two reasons. First, the stabilizer in the case χ is SL(2) versus C * for the generic point of σ. Secondly, the singularities along the rational normal curve are of transversal type A 2 for χ versus transversal type A 1 for a general point on σ.
6.3. Cubic fourfolds singular along a non-degenerate curve. In order to establish theorem 6.1, the last case we have to consider is that of cubic fourfolds singular along a non-degenerate curve. We prove that in this case the cubic is determinantal (see equation (6.2)) and generically stable. The first step for this is a geometric argument showing that there are only two possible types of nondegenerate curves that occur in the singular locus of a cubic fourfold.
Proposition 6.9. Let Y be a cubic fourfold with dim Sing(Y ) = 1. Assume that C ⊂ Sing(Y ) is an irreducible non-degenerate curve. Then C is either a rational normal quintic or an elliptic normal sextic or degeneration of it.
Proof. Let C ⊆ Sing(Y ) be an irreducible non-degenerate curve of degree d. We choose a generic point p ∈ C and project Y onto a generic hyperplane H ∼ = P 4 . Inside H we obtain a sextic surface S 1 (a (2, 3) complete intersection) that parametrizes the lines passing through p. The projection C 1 of C is irreducible, non-degenerate, birational to C ([6, pg. 109]), and of degree d − 1 ([13, pg. 235]). Since Y is singular along C, the surface S 1 contains C 1 . It is easy to see that S 1 is singular along C 1 . Note that the surface S 1 is non-degenerate and reduced (otherwise dim Sing(Y ) ≥ 2). For degree reasons, the only possibility for S 1 to be reducible is to be the union of two non-degenerate cubic scrolls. We repeat the construction with C 1 and S 1 . By projecting from a generic point of C 1 we obtain a curve C 2 and a surface S 2 . The curve C 2 is birational to C and of degree d−2 in P 3 . Since we project from a general point of C 1 , and S 1 is a complete intersection, one checks that S 1 maps birationally onto its image. It is also clear that C 2 ⊂ Sing(S 2 ). Since we projected from a singular point of S 1 , the surface S 2 will have degree 4. We have two cases: either S 2 is irreducible or not. If the quartic S 2 is irreducible, then the degree of C 2 is at most 3 (a generic hyperplane section is an irreducible plane quartic, which has at most 3 singular points). In this case, we conclude that C 2 is the twisted cubic. It follows that C is a rational normal quintic in P 5 . On the other hand, if S 2 is reducible, the only non-degenerate case is of C 2 being a component of a complete intersection of two quadrics. Thus C 2 has degree 3 or 4 and is rational or elliptic. The conclusion follows.
We have the following dimension count for the space of cubic fourfolds singular along a given elliptic sextic curve or rational normal curve: Lemma 6.10. Let C be an elliptic normal curve of degree 6 (or rational normal curve of degree 5) in P 5 . Then, the linear system of cubics singular along C is 1-dimensional (resp. 3-dimensional).
Proof. The cubics singular along C correspond to the global sections of I 2 C (3). Starting from the exact sequences: and 0 → I 2 C → I C → I C /I 2 C → 0, one obtains (via a standard Riemann-Roch computation) the Euler characteristic: where n = 5 is the dimension of the projective space, d and g are the degree and genus of C. In the rational case, we have the expected dimension χ(I 2 C (3)) = 4. Similarly, if C is an elliptic sextic, then χ(I 2 C (3)) = 2. To compute the actual dimensions of H 0 (I 2 C (3)), we combine the Euler characteristic computation with some vanishing results. Specifically, by a result of Rathmann (see [27,Thm. 1.1]) one is guaranteed the vanishing of H i (P n , I 2 C (k)) = 0 for all k ≥ 3 and i > 0, provided that the degree of the curve is high with respect to the genus, e.g. d ≥ 2g + 3 suffices. In both cases considered here, the vanishing result applies. Thus, h 0 (I 2 C (3)) = χ(I 2 C (3)) and the lemma follows. The structure of the cubic fourfolds singular along an elliptic normal curve is described by the following result: Lemma 6.11. Let Y be a cubic fourfold whose equation is given by the determinant of a 3 × 3 matrix A of linear forms (we call Y determinantal). Then Y is singular along the curve cut by the minors of A. The singular locus of a general determinantal cubic fourfold is precisely an elliptic normal curve of degree 6. Conversely, a cubic fourfold singular along an elliptic normal curve of degree 6 is determinantal.
Proof. The fourfold Y is determinantal if and only if it is a linear section L of the secant variety (a cubic hypersurface in P 8 ) of the Segre fourfold (the image of P 2 × P 2 ֒→ P 8 ). In the generic case the singular locus of Y is the intersection of the section L with the Segre fourfold. The claim now follows from standard facts on the Segre embedding. Conversely, the fact that a cubic fourfold singular along an elliptic sextic is determinantal is a classical result, e.g. Room [24, §7.11] (see also [12,Ex. 2.11]). Remark 6.12. According to lemma 6.10, there exists a pencil of cubic fourfolds singular along a given elliptic normal curve E ֒→ P 5 . We note that this pencil is not trivial: there are 4 special members of the pencil where the corresponding cubic is singular along a Veronese surface. Namely, the elliptic curve E is embedded in P 5 by a complete linear system |D| of degree 6. Let D ′ ∈ Pic(E) such that D = 2D ′ . It follows that the embedding of E in P 5 can be factored as a composition of an embedding of E in P 2 (given by |D ′ |) followed by a Veronese embedding P 2 ֒→ P 5 . Thus, E sits on a Veronese surface S. Then, Sec(S) is singular along S (and thus E) and gives a section of I 2 E (3). Since D ′ is defined up to addition of points of order 2 in the Jacobian of E, we obtain 4 such sections.
We now note that the rational curve case is a specialization of the elliptic case. Lemma 6.13. A cubic fourfold singular along a rational normal curve of degree 5 is determinantal. In particular, Y is also singular along a line.
Proof. Let C be a rational normal curve of degree 5. Without loss of generality, we can assume that C is defined by the 2 × 2 minors of the matrix: Since the secant variety of C is cut by the 3 × 3 minors of M ([13, Prop. 9.7]), it follows that the four 3 × 3 minors of M give four linearly independent sections of I 2 C (3). By lemma 6.10, we have dim H 0 (P 5 , I 2 C (3)) = 4. Thus, any section of I 2 C (3) is a linear combination of the minors of M . In conclusion, a cubic fourfold singular along a rational normal curve has the equation: for a, b, c, d ∈ C, which can be then arranged in a determinantal form.
We conclude: Proposition 6.14. Let Y be an irreducible cubic fourfold singular locus along a non-degenerate curve C. Then Y is determinantal (i.e. given by (6.2)). A general determinantal cubic is stable. The locus of determinantal cubics in M forms an irreducible surface φ, which meets the other components of M \ M in the curve τ .
Proof. We can restrict to the case dim Sing(Y ) = 1 (otherwise it is easily checked that Y is determinantal). By proposition 6.9, Y is singular along a (non-degenerate) rational normal curve or an elliptic normal curve. According to lemmas 6.11 and 6.13, in either of these cases Y is determinantal. If Y is general, then Y is singular along a non-degenerate elliptic normal curve with A ∞ singularities. By theorem 3.2, it follows that Y is stable. We define φ to be the closure in M of the locus of such fourfolds. Clearly, φ is 2-dimensional: one dimension for the modulus of the elliptic curve, and one dimension for the choice of a point in the pencil of cubics singular along a given elliptic curve (see 6.12).
The cubics parameterized by the curve τ (including the special points ζ and ω) are determinantal. It follows that τ is included in the surface φ. The remaining part of the proposition follows from two observations. First, if Sing(Y ) degenerates to a union of two or more curves, then, for degree reasons, Y is not stable (cf. theorem 3.2). Finally, if Y is strictly semistable and determinantal, the minimal orbit contained in the orbit of Y has to be of type τ (again for degree reasons).

Proof of the main results
At this point, the main theorems 1.1 and 1.2 are obtained as simple consequences of the results proved in the previous section. For reader convenience, we collect the essential information about the boundary strata in table 3. We recall that the boundary strata α-φ are closed. The various degeneracy loci inside these boundary strata (mostly corresponding to intersections of two or more strata) are given in table 4 (see also figure 2).

Stratum Equations
Discussion Singularities Dim.  The degeneracy conditions corresponding to cases S1, S2, and S5 are equivalent to saying that Y is singular along a curve contained in a 3-dimensional linear subspace of P 5 (cf. proposition 6.4). The conditions S3, S4, and S6 are equivalent to requiring that Y contains a singular point p that deforms to a singularity of E r for r = 8, 7, or 6 respectively (see the discussion of section 5, especially theorem 5.6). We note that the proofs of section 5 apply also to the case when the singularity at p is non-isolated. The boundary components of the GIT compactification M s ⊂ M parametrize the minimal orbits of strictly semi-stable cubic fourfolds. According to theorem 3.2, the strictly semistable cubic fourfolds with minimal orbits are of type α-δ, producing the boundary strata α-δ. The detailed analysis of these cases is done in section 4. Finally, the points in M s \ M correspond to the orbits of stable cubic fourfolds with non-isolated singularities. By theorem 6.1, the only relevant cases are those of the cubic fourfolds singular in a rational normal curve of degree 4 or an elliptic normal curve of degree 6. These two cases produce the boundary strata ǫ and φ (see §6.2 and §6.3). For the discussion of the possible degeneracies and adjacencies we refer to the respective sections (see table 3).

Further remarks
8.1. Relation to the stability of cubic threefolds. The analysis of stability for cubic fourfolds is a natural extension of the similar analysis for cubic threefolds of Allcock [1] and Yokoyama [31]. Thus, there is no surprise that our results parallel to a large extent the results for cubic threefolds (compare for instance theorem 3.2 to [1, Thm. 1.3]). Here, we comment on a more direct link between the results for cubic threefolds and fourfolds.
We recall that the construction of Allcock-Carlson-Toledo [4] associates to a cubic threefold X ⊂ P 4 the cubic fourfold Y obtained as a triple cyclic cover of P 4 branched along X. It is not hard to see (due to the µ 3 stabilizer) that the threefold X gives a closed orbit if and only if the fourfold Y does. This fact helped our analysis in two ways. First, the singularities allowed for stable cubic threefolds (i.e. A 1 , . . . , A 4 ) are precisely those that give stable singularities for the associated cubic fourfold Y . Thus, [1, Theorem 1.1] anticipates our result on the stability of cubic fourfolds with simple singularities. Similarly, the boundary strata for cubic threefolds should correspond to boundary strata for cubic fourfolds. Indeed, this is the case: the strata T and ∆ of [1, Thm. 1.2] correspond to our strata β and δ respectively. Furthermore, the special point of T corresponding to the chordal cubic corresponds to the special stratum χ ⊂ β. Roughly speaking, the strata β and δ are generated by T and ∆ in a natural way. For example, δ can be interpreted as a j-line with ∆ corresponding to the elliptic curve of j-invariant 0. Thus, δ is obtained from ∆ by allowing the j-invariant to vary. 8.2. Relation to the stability of plane sextics. Hassett [14, §4.4] has noticed that the point ω corresponding to the secant variety of the Veronese surface plays a special role in the Hodge theoretical analysis of cubic fourfolds. Specifically, as one degenerates to ω, one obtains as the essential part of the limit mixed Hodge structure the Hodge structure of a degree 2 K3 surface. Geometrically, this is explained by the following observation. Let F 0 be the equation of the secant to the Veronese surface S ∼ = P 2 in P 5 . Consider a general pencil of cubics (F 0 + tF ) t∈P 1 degenerating to F 0 . Then, the base locus of the pencil cuts on the Veronese surface a plane sextic curve C. The degree 2 K3 surface mentioned above is the double cover of P 2 along the sextic C (the relation of Hodge structures is then obtained by applying the Clemens-Schmid exact sequence).
By interpreting in a more intrinsic way the observation mentioned above, we find a natural relationship between the stability of cubic fourfolds and the stability of plane sextics (analyzed by Shah [25]). Namely, locally near ω (in theétale topology) M is the quotient of an equivariant slice to the orbit ω by the stabilizer subgroup (Luna's slice Theorem [21,Appendix D]). The stabilizer corresponding to ω is a subgroup H ∼ = SL(3) of G ∼ = SL (6). The normal slice to the orbit of ω is computed as follows. Let W = H 0 (P 5 , O P 5 (1)) ∼ = H 0 (P 2 , O P 2 (2)) (isomorphism induced by the Veronese embedding). It follows that W ∼ = Sym 2 V as an SL(3)-representation (where V is the standard SL(3)-representation) and then (8.1) Sym 3 (W ) ∼ = Sym 3 (Sym 2 V ) ∼ = C ⊕ Γ 2,2 ⊕ Sym 6 V (cf. [10, (13.15)]). It is easy to see that the trivial summand in (8.1) corresponds to the cubic fourfolds containing a fixed Veronese surface. Similarly, the Γ 2,2 summand corresponds to the tangent directions to the orbit ω. Thus, the SL(3)-representation on the normal space is Sym 6 V . We conclude that locally near ω the space M is isomorphic to Sym 6 V /SL(3), i.e. the affine cone over the GIT quotient for plane sextics. In particular, it follows that near ω the boundary structure of M is essentially the same as that for plane sextics. Thus, we have the following natural correspondence between the boundary components adjacent to ω and the list of Shah for plane sextics (see [25,Thm. 2.4]): (Type II) β, γ, ǫ, and φ correspond to II(1), II(2), II(3), and II(4) respectively; (Type III) σ, ζ correspond to III(1), and III(2) respectively; (Type IV) χ corresponds to IV; (N.B. a geometric interpretation of this matching is easily seen by considering pencils of cubics degenerating to the secant to the Veronese, as explained above).
8.3. Stratification of the boundary. As mentioned above, the link between cubic fourfolds and degree 6 plane curves is both at the level of GIT and Hodge theory. This motivates us to introduce a natural stratification of the boundary of the moduli space of cubic fourfolds. The stratification is defined in terms of the complexity (measured in "types") of the singularities for semistable cubic fourfolds.
The stratification is meaningful from a GIT point of view, but, as in the case of Shah [25], the real reason for introducing it is Hodge theoretic.
Definition 8.1. We say Y is a type I cubic fourfold if Y is smooth or has at worst simple isolated singularities. We say Y is a type II cubic fourfold if Y is not of type I, and all the singularities of Y are of the following types: (0) isolated simple singularities; (1) isolated singularities of type E r (r = 6, 7, 8); (2) non-isolated singularities of type A ∞ (locally a double line); (3) non-isolated singularities of type D ∞ (simple pinch point).
Our results say that a cubic fourfold of type I is stable, and that one of type II is semi-stable. A generic boundary point belonging to a strata α, . . . , φ is of type II. The boundary locus that does not parameterize cubic fourfolds of type II is precisely the surface σ. Thus, we define: Definition 8.2. Let Y be a semi-stable cubic fourfold with minimal orbit. We say that Y is of type III if it is of type σ, but not of type χ. We say that Y is of type IV if it is of type χ.
Remark 8.3. The type IV fourfolds are the "most degenerate" fourfolds that occur in the GIT analysis. This statement has meaning both in terms of singularities and in terms of GIT (see 4.2 and 6.8). A type IV has non-isolated singularities of transversal type A 2 (versus A 1 for type III). Also, the type IV fourfolds are the only semi-stable cubic fourfolds that are stabilized by a semisimple algebraic group.
In conclusion, we have stratified the boundary M \ M = α ∪ . . . φ in type II boundary: the open part (α ∪ φ) \ σ, type III boundary: the surface σ \ χ, and type IV boundary: the curve χ. This stratification is completely analogous to the stratification of Shah [25] for plane sextics. The types correspond to certain monodromy properties of the degenerations to a boundary point. Here, we only mention a geometric peculiarity that plays an important role for the monodromy analysis: to each type II fourfold there is associated in a natural way an elliptic curve, or more precisely a j-invariant. Namely, a type II cubic is singular along an elliptic curve, along a rational curve with 4 special points, or it has an E r singularity (see the various lemmas where the boundary strata are analyzed); in each case the meaning of the j-invariant is clear. The type III fourfolds correspond to the case of j-invariant equal to ∞. For type IV fourfolds, one can not associate a meaningful j-invariant. Roughly speaking, the associated elliptic curve becomes singular for the type III case, and completely vanishes for type IV.