STABILITY OF ASSOCIATED FORMS

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1 STABILITY OF ASSOCIATED FORMS MAKSYM FEDORCHUK AND ALEXANDER ISAEV Abstract. We show that the associated form, or, equivalently, a Macaulay inverse system, of an Artinian complete intersection of type (d,..., d) is polystable. As an application, we obtain an invariant-theoretic variant of the Mather-Yau theorem for homogeneous hypersurface singularities. Contents 1. Introduction 1 2. Associated forms of complete intersections 4 3. Recognition criterion for decomposable balanced complete intersections 8 4. Preservation of polystability Invariant-theoretic variant of the Mather-Yau theorem 16 References Introduction In this paper, we establish the GIT polystability of Macaulay inverse systems for Gorenstein Artin algebras given by balanced complete intersections. This leads to a purely invariant-theoretic solution to the problem of deciding when two such algebras are isomorphic. An important example of a balanced complete intersection is the Milnor algebra of an isolated homogeneous hypersurface singularity and so, as an application of our polystability result, we obtain an algebraic variant of the famous Mather-Yau theorem for such singularities over an arbitrary field of characteristic zero. We will now explain our approach. Recall that for two homogeneous forms f(x 1,..., x n ) and g(x 1,..., x n ), the problem of determining whether one can be obtained from the other by a linear change of variables can often be solved by a purely algebraic method offered by Geometric Invariant Theory (GIT). Namely, if deg f = deg g = d, then the question can be rephrased as whether we have the equality of the orbits GL(n) f = GL(n) g Mathematics Subject Classification: 14L24, 13A50, 13H10, 32S25 Keywords: complete intersections, Geometric Invariant Theory, associated forms, isolated hypersurface singularities. 1

2 2 MAKSYM FEDORCHUK AND ALEXANDER ISAEV under the natural action of GL(n) on Sym d V, where V is the standard representation of GL(n). When f and g are polystable in the sense of GIT, their orbits in Sym d V can be distinguished using invariants. Namely, the two orbits are distinct if and only if there exists a relative GL(n)-invariant I on Sym d V, that is, an SL(n)- invariant homogeneous element of Sym ( Sym d V ), such that I(f) = 0 and I(g) 0. Furthermore, the famous Gordan-Hilbert theorem implies that we can find finitely many relative GL(n)-invariants I 1,..., I N of equal weights such that for polystable f and g, we have (1.1) GL(n) f = GL(n) g [I 1 (f) : : I N (f)] = [I 1 (g) : : I N (g)]. One can consider a generalization of the above question and ask when two m- dimensional linear systems f 1,..., f m and g 1,..., g m of degree d forms are related by a linear change of variables. This again can be phrased in terms of a GIT problem, this time given by the action of SL(n) on Grass(m, Sym d V ) (or the SL(n)-action on the affine cone over the Grassmannian in its Plücker embedding). A priori, to distinguish orbits of this action, one needs to understand polynomial SL(n)- invariants on m Sym d V. One of the main results of this paper is that for m = n, i.e., when the number of forms is equal to the number of variables, the problem of distinguishing SL(n)-orbits in Grass(n, Sym d V ) can often be reduced to that of distinguishing SL(n)-orbits of degree n(d 1) forms in n (dual!) variables. The reason behind this simplification is that a generic n-dimensional subspace U Sym d V is spanned by a regular sequence g 1,..., g n. The algebra A := k[x 1,..., x n ]/(g 1,..., g n ) is then a graded local Gorenstein Artin algebra of socle degree n(d 1). The homogeneous Macaulay inverse system of this algebra is then an element of P Sym n(d 1) V, which we call the associated form of U. A classical theorem of Macaulay says that the associated form morphism A sending U to its associated form is injective (see [18], [13, Lemma 2.12], or [10, Exercise 21.7]). Alper and Isaev, who initiated a systematic study of this morphism, showed that A is a locally closed immersion, and conjectured that A preserves GIT semistability and that the induced morphism on the GIT quotients is also a locally closed immersion; we refer the reader to [2] for details, for the motivation behind these conjectures, and for a proof in the case of binary forms. In [12], the first author proved that A preserves GIT semistability. Here we show that A preserves GIT polystability (see Theorem 2.6) and that, consequently, injectivity is preserved on the level of the GIT quotients, just as Alper and Isaev conjectured. As an application of our result, for isolated homogeneous hypersurface singularities, we obtain an algebraic variant of the Mather-Yau theorem (see [19]). This theorem states that an isolated hypersurface singularity in C n is determined, up to biholomorphism, by n and the isomorphism class of its moduli algebra. The Mather- Yau theorem is nontrivial even for homogeneous singularities and raises a natural

3 STABILITY OF ASSOCIATED FORMS 3 question of how exactly a singularity is encoded by the corresponding algebra (see [5]). If f(x 1,..., x n ) = 0 is such a singularity, defined by a form of degree d + 1, then its moduli algebra coincides with its Milnor algebra M f, which has an associated form A(f) Sym n(d 1) V, first studied in [1]. Our polystability result implies that two forms f, g Sym d+1 V define isomorphic isolated hypersurface singularities (i.e., the completions of the local rings of the hypersurfaces {f = 0} and {g = 0} at the origin are isomorphic over the algebraic closure of the field) if and only if their associated forms A(f) and A(g) map to the same point in the GIT quotient P Sym n(d 1) V // SL(n), something that can be detected by finitely many relative GL(n)-invariants just as in (1.1). Since the associated form A(f) is computable from the Milnor algebra alone, we obtain a purely algebraic, and in principle algorithmic, way of deciding when two isolated homogeneous hypersurface singularities are isomorphic based solely on their Milnor algebras (see Theorem 5.2). Notation and conventions. We work over a field k of characteristic 0 (not necessarily algebraically closed). The dual of a k-vector space will be denoted by. Fix n 1 and let V be an n-dimensional vector space over k. Let S := Sym V be the symmetric algebra on V with the standard grading. We briefly recall some basic notions of GIT, as developed in [20], that are used in this paper. Suppose W is an algebraic representation of a reductive group G. Then a vector x W is semistable if 0 / G x and polystable if x is semistable and G x is closed. Similarly, for x PW, we say that x is semistable (resp., polystable) if some (equivalently, any) lift x of x to W is semistable (resp., polystable). The locus of semistable points in PW is open and is denoted by PW ss. More generally, if X PW is a G-invariant projective closed subscheme, then one defines the locus of semistable points in X as X ss := X PW ss. The orbits of polystable k-points in X ss are in bijection with the k-points of the projective GIT quotient X ss //G := Proj m 0 H 0( X, O X (m) ) G. In particular, polystable orbits in PW ss (and more generally in X ss ) are distinguished by G-invariant forms on PW. It will be crucial for us that in the case of a perfect field, semistability and polystability is determined by the standard Hilbert- Mumford numerical criterion; see [16] for more details. Roadmap of the paper. In Section 2, we introduce the main actors of this work, the balanced complete intersection algebras and their associated forms, and state our principal result (Theorem 2.6). In Section 3, we prove a key technical commutative algebra proposition. In Section 4, we prove Theorem 2.6. Finally, in Section 5, we give applications of our preservation of polystability result, the main of which is an invariant-theoretic variant of the Mather-Yau theorem.

4 4 MAKSYM FEDORCHUK AND ALEXANDER ISAEV Acknowledgement. During the preparation of this work, the first author was supported by the NSA Young Investigator grant H and Alfred P. Sloan Research Fellowship, and the second author was supported by the Australian Research Council. 2. Associated forms of complete intersections 2.1. Gorenstein Artin algebras and Macaulay inverse systems. Recall that a homogeneous ideal I S is Gorenstein if A := S/I is a Gorenstein Artin k- algebra, meaning that dim k A < and dim k Soc(A) = 1. Here, Soc(A) is the annihilator of the unique maximal ideal m A of A. We endow A with the standard grading coming from S. Then ν A = A d, d=0 where ν is the socle degree of A, and Soc(A) = A ν. The surjection H ν : S ν A ν is called the ν th Hilbert point of A, which we regard as a point in PS ν. As we will see shortly, it is dual to the homogeneous Macaulay inverse system of A. A key way to interpret the fact that A is Gorenstein is via the following standard lemma: Lemma 2.1. A homogeneous ideal I S defines a Gorenstein Artin k-algebra A = S/I of socle degree ν if and only if codim(i ν, S ν ) = 1, I d = S d for every d > ν, and for H ν : S ν A ν and every d < ν we have I d = { ( ) } f S d H ν (gf)ν = 0, for all g S (2.2) = { } f S d fg I ν for all g S ν d. Remark 2.3. Equality (2.2) is equivalent to the fact that the pairing A d A ν d A ν is perfect for every d < ν. We can regard S = Sym V as a ring of polynomial differential operators on a dual ring D := Sym V as follows. Let x 1,..., x n be a basis of V and z 1,..., z n be the dual basis of V. Then we have an apolarity action of S on D given by differentiation : S D D f(x 1,..., x n ) g(z 1,..., z n ) := f( / z 1,..., / z n )g(z 1,..., z n ). Since char(k) = 0, the restricted pairing S d D d k is perfect and so defines an isomorphism (2.4) D d S d. Let I S be a Gorenstein ideal and ν the socle degree of the algebra A = S/I. Recall that a (homogeneous) Macaulay inverse system of A is an element f D ν

5 such that STABILITY OF ASSOCIATED FORMS 5 f = I (see [13, Lemma 2.12] or [10, Exercise 21.7]). Clearly, the line f PD ν maps to the ν th Hilbert point H ν PSν of A under the isomorphism (2.4) for d = ν and all Macaulay inverse systems are mutually proportional. For any ω Sν with ker ω = I ν, papers [8, 9] introduced an associated form of A as the element of D ν given by the formula (2.5) f A,ω := ω ( (x 1 z x n z n ) ν) k[z 1,..., z n ] ν. Since fa,ω = I, these associated forms give explicit formulae for the Macaulay inverse systems of A (see [14] for more details) Koszul complex. Suppose m is a positive integer. Recall that for g 1,..., g m S d, the Koszul complex K (g 1,..., g m ) is defined as follows. Let e 1,..., e m be the standard degree d generators of the graded free S-module S( d) m. Then K (g 1,..., g m ) is an (m + 1)-term complex of graded free S-modules with K j (g 1,..., g m ) := j S( d) m for j = 1,..., m, K 0 (g 1,..., g m ) := S, and the differential d j : K j (g 1,..., g m ) K j 1 (g 1,..., g m ) given by d j (e i1 e ij ) := j ( 1) k 1 g ir e i1 ê ir e ij. r=1 Note that H 0 (K (g 1,..., g m )) = S/(g 1,..., g m ) Balanced complete intersections and their associated forms. Suppose d 2 and m n. Recall that elements g 1,..., g m S d form a regular sequence in S if and only if any of the following equivalent conditions hold: (1) codim(g 1,..., g m ) = m. (2) the Koszul complex K (g 1,..., g m ) is a minimal free resolution of S/(g 1,..., g m ), Moreover, if n = m, then the above conditions are also equivalent to each of (3) the forms g 1,..., g n have no nontrivial common zero in k n. (4) the resultant Res(g 1,..., g n ) is non-zero. We now recall the definition of the associated form of a complete intersection as first given in [2]. If g 1,..., g n S d form a regular sequence in S, then we call I := (g 1,..., g n ) a balanced complete intersection ideal of type (d) n and A := S/I a balanced complete intersection algebra of type (d) n. A balanced complete intersection algebra of type (d) n is a graded Gorenstein Artin k-algebra with Hilbert function ( ) 1 t dim k (A j )t j d n = 1 t j 0 and so has socle degree n(d 1).

6 6 MAKSYM FEDORCHUK AND ALEXANDER ISAEV Let H n(d 1) : S n(d 1) A n(d 1) be the n(d 1) th Hilbert point of A. Denote by Jac(g 1,..., g n ) the Jacobian n n matrix of g 1,..., g n, whose (ij) th entry is g i / x j. Then A n(d 1) is spanned by Jac := H n(d 1) (det Jac(g 1,..., g n )) (see [21, p. 187]). It follows that we can choose an isomorphism A n(d 1) k that sends Jac to 1. Denote the resulting element of S n(d 1) = Hom k(s n(d 1), k) by ω. Then the form f A,ω D n(d 1) given by Equation (2.5) is called the associated form of g 1,..., g n and is denoted by A(g 1,..., g n ) (cf. [2]). We let Grass(n, S d ) Res be the affine open subset of Grass(n, S d ) on which the resultant (considered as a section of the corresponding line bundle) does not vanish. Alper and Isaev defined the associated form morphism A: Grass(n, S d ) Res PD n(d 1), that sends a point U Grass(n, S d ) Res to the line spanned by A(g 1,..., g n ), where g 1,..., g n is any basis of U (see [2, Section 2]). By [2, Lemma 2.7], the morphism A is SL(n)-equivariant. The preservation of GIT polystability by A is the main object of study in this paper, and the following is our main result: Theorem 2.6. Suppose U Grass(n, S d ) Res is polystable. Then A(U) is polystable. While proving Theorem 2.6, we also simplify the proof of the semistability of associated forms, first obtained by the first author in [12, Theorem 1.2]. We refer the reader to Theorem 4.6 for a more technical version of Theorem 2.6 that gives a necessary and sufficient condition for A(U) to be stable when k is algebraically closed Balanced complete intersections and decomposability. Among all codimension n ideals of S generated in degree d, the balanced complete intersections are distinguished using the following simple, but important lemma: Lemma 2.7. Suppose J S is a codimension n homogeneous ideal generated by J d. Then either J is a balanced complete intersection or (S/J) n(d 1) = 0. Proof. Since J has codimension n in S, there exist r 1,..., r n J d that form a regular sequence. Set Y := (r 1,..., r n ) J. Then S/Y is a balanced complete intersection algebra with socle in degree n(d 1). Let H n(d 1) be the n(d 1) th Hilbert point of S/Y. We have two possibilities: (1) either H n(d 1) ((s) n(d 1) ) = 0 for every s J d, in which case J = Y, (2) or there exists s J d such that H n(d 1) ((s) n(d 1) ) 0. In the latter case, (s) n(d 1) Y n(d 1), so that J n(d 1) strictly contains Y n(d 1). Since Y n(d 1) is already of codimension 1 in S n(d 1), we conclude that J n(d 1) = S n(d 1) in this case. Definition 2.8. We say that U Grass(n, S d ) Res is decomposable if there is a choice of a basis x 1,..., x n of S 1, an integer 1 a n 1, and a basis g 1,..., g n of U such that g a+1,..., g n k[x a+1,..., x n ]. An element U Grass(n, S d ) Res that is

7 STABILITY OF ASSOCIATED FORMS 7 not decomposable will be called indecomposable. For U Grass(n, S d ) Res, we will also speak about the (in)decomposability of the balanced complete intersection ideal I := (U) S and the balanced complete intersection algebra S/I. Decomposable complete intersections have a simple structure described by the following result: Proposition 2.9. Suppose U = g 1,..., g n Grass(n, S d ) Res is such that, for some 1 a n 1, we have g a+1,..., g n k[x a+1,..., x n ] d. Then g a+1,..., g n is a regular sequence in k[x a+1,..., x n ] d and there exists a regular sequence g 1,..., g a k[x 1,..., x a ] d such that the closure of the SL(n)-orbit of U Grass(n, S d ) Res contains g 1,..., g a, g a+1,..., g n. Proof. The first statement is clear. For the second, set g i := g i (x 1,..., x a, 0,..., 0). Then g 1,..., g a form a regular sequence in k[x 1,..., x a ]. Let µ be the 1-PS of SL(n) acting with weight (n a) on x 1,..., x a and with weight a on x a+1,..., x n. Then lim µ(t) U = g t 0 1,..., g a, g a+1,..., g n Grass(n, S d ) Res. This finishes the proof. The following is immediate: Corollary Suppose a decomposable U Grass(n, S d ) Res is polystable. Then there exists 1 a n 1 and a basis x 1,..., x n of S 1 such that U = g 1,..., g a, g a+1,..., g n, where g 1,..., g a is a regular sequence in k[x 1,..., x a ] d and g a+1,..., g n is a regular sequence in k[x a+1,..., x n ] d. The balanced complete intersections described by the previous corollary are called direct sums. Their associated forms are computed as follows. Lemma Suppose for 1 a n 1, we have that g 1,..., g a k[x 1,..., x a ] d is a regular sequence and g a+1,..., g n k[x a+1,..., x n ] d is a regular sequence. Then ( ) n(d 1) A(g 1,..., g n ) = A(g 1,..., g a )A(g a+1,..., g n ), a(d 1) where A(g 1,..., g a ) is the associated form of g 1,..., g a in k[z 1,..., z a ] a(d 1), and A(g a+1,..., g n ) is the associated form of g a+1,..., g a in k[z a+1,..., z n ] (n a)(d 1). Proof. As an element of S n(d 1), the associated form A(g 1,..., g n ) is uniquely determined by the property that for every f S n(d 1) (or, equivalently, for every monomial f S n(d 1) ), we have (2.12) f 1 (n(d 1))! A(g 1,..., g n )(f) ( det Jac(g 1,..., g n ) ) (g 1,..., g n ) n(d 1). It remains to observe that the following three statements

8 8 MAKSYM FEDORCHUK AND ALEXANDER ISAEV 1 (1) f A(g (a(d 1))! 1,..., g a )(f) ( det Jac(g 1,..., g a ) ) (g 1,..., g a ) for every f k[x 1,..., x a ] a(d 1) and k[x 1,..., x a ] N (g 1,..., g a ) N for every N > a(d 1), 1 (2) f A(g ((n a)(d 1))! a+1,..., g n )(f) ( det Jac(g a+1,..., g n ) ) (g a+1,..., g n ) for every f k[x a+1,..., x n ] (n a)(d 1) and k[x a+1,..., x n ] N (g a+1,..., g n ) N for every N > (n a)(d 1), (3) det Jac(g 1,..., g n ) = det Jac(g 1,..., g a ) det Jac(g a+1,..., g n ) imply that for every monomial f = f 1 f 2 S n(d 1), where f 1 k[x 1,..., x a ] a(d 1) and f 2 k[x a+1,..., x n ] (n a)(d 1), we have that ( ) 1 n(d 1) f A(g 1,..., g a )(f 1 )A(g a+1,..., g n )(f 2 ) ( det Jac(g 1,..., g n ) ) (n(d 1))! a(d 1) lies in (g 1,..., g n ) n(d 1) and that f (g 1,..., g n ) n(d 1) for any other monomial in S n(d 1). The lemma now follows by appealing to (2.12). 3. Recognition criterion for decomposable balanced complete intersections Decomposable balanced complete intersections play a crucial role in our inductive proof of polystability of associated forms. In this section, we obtain a criterion for a balanced complete intersection ideal to be decomposable based only on partial information about the ideal. Although technical, this result may be of independent interest; in fact, it has already been used by the first author to give a new criterion for smooth forms to be of Sebastiani-Thom type [11]. We note that the results of this section are valid over an arbitrary field k, with no restriction on its characteristic. Proposition 3.1. Let 1 b n 1 and suppose that I k[x 1,..., x n ] is a balanced complete intersection ideal of type (d,..., d) such that (A) The homomorphic image of I in k[x 1,..., x n ]/(x b+1,..., x n ) k[x 1,..., x b ] is a balanced complete intersection ideal. Equivalently, dim k ( Id (x b+1,..., x n ) ) = n b. (B) (x b+1,..., x n ) (n b)(d 1)+1 I (n b)(d 1)+1. Then there exist n b linearly independent elements g b+1,..., g n I d k[x b+1,..., x n ]; in particular, I is decomposable. Remark 3.2. Note that conditions (A) and (B) are necessary for the conclusion to hold. Indeed, if g b+1,..., g n I d k[x b+1,..., x n ] is a regular sequence, then I d (x b+1,..., x n ) = (g b+1,..., g n ), and (g b+1,..., g n ) is a balanced complete intersection ideal in k[x b+1,..., x n ] and so contains (x b+1,..., x n ) (n b)(d 1)+1. The difficulty of proving Proposition 3.1 lies in verifying the sufficiency of these conditions. Proof of Proposition 3.1. Note that for b = n 1, Proposition 3.1 is obvious, because in this case x d n I d by (B). So we assume that b n 2 in what follows.

9 STABILITY OF ASSOCIATED FORMS 9 Using condition (A) we can find a basis g 1,..., g n of I d such that g b+1,..., g n (x b+1,..., x n ). We will prove that in fact g b+1,..., g n k[x b+1,..., x n ]. We separate our argument into two parts, given by Lemmas 3.3 and 3.9. Lemma 3.3. Suppose g 1,..., g n is a balanced complete intersection ideal such that g b+1,..., g n (x b+1,..., x n ) and condition (B) of Proposition 3.1 holds. Then (x b+1,..., x n ) (n b)(d 1)+1 (g b+1,..., g n ). Proof of Lemma 3.3. Set N := (n b)(d 1) + 1, and let J := (x b+1,..., x n ) N. By assumption, we have J I. Let J := J k[x b+1,..., x n ] and R := k[x b+1,..., x n ]. Then J is the N th power of the irrelevant ideal in R and so has regularity N, for example, by the computation of its local cohomology (cf. [4, Lemma 1.7]). Hence J has a linear minimal free resolution 1 as an R-module. In other words, the minimal free resolution of J has the following form: (3.4) 0 R l n b ( N n + b + 1) R l 2 ( N 1) R l 1 ( N) J 0. Since S is a flat R-algebra, tensoring by S we obtain a minimal free resolution of J as an S-module: 0 S l n b ( N n + b + 1) S l 2 ( N 1) S l 1 ( N) J 0. Consider now the Koszul complex K (g 1,..., g n ), which gives a minimal free resolution of S/I. The inclusion J I gives rise to a map of complexes (3.5) S l n b ( N n + b + 1) S l 1 ( N) S m n b K n b (g 1,..., g n ) K 1 (g 1,..., g n ) S. m 1 m 0 =Id S Next, note that the Koszul complex K (g 1,..., g n ) contains K (g b+1,..., g n ) as a subcomplex. Let Q be the quotient complex. Then Q i := K i (g 1,..., g n )/K i (g b+1,..., g n ) is a free S-module for every i 0, Q 0 = 0, H 1 (Q ) = I/(g b+1,..., g n ) S/(g b+1,..., g n ), and H i (Q ) = 0 for i > 1. Composing (3.5) with the quotient morphism, and replacing Q 0 by S/(g b+1,..., g n ), we obtain a map of exact complexes (3.6) S l n b ( N n + b + 1) S l 1 ( N) S m n b Q n b Q 1 S/(g b+1,..., g n ). m 1 m 0 1 As explained in [6, pp ], an explicit minimal free resolution of J was constructed by Buchsbaum and Rim using the Eagon-Northcott complex [7].

10 10 MAKSYM FEDORCHUK AND ALEXANDER ISAEV Note that J (g b+1,..., g n ) if and only if m 0 (J) = Im(d 1 m 1 ) = 0, which is what we are going to prove. We begin with the following: Claim 3.7. m n b = 0. Proof. The key observation is that N n + b + 1 = (n b)d, and K n b (g 1,..., g n ) S ( n n b) ( (n b)d ). This implies that m n b in (3.5) is given by a matrix of scalars. Next, we have a direct sum decomposition where K n b (g 1,..., g n ) = K n b (g b+1,..., g n ) Q n b, Q n b = i 1 < <i n b, i 1 b S( (n b)d)e i1 e in b. We need to prove that Im(m n b ) K n b (g b+1,..., g n ). Note that after setting x b+1 = = x n = 0, all the differentials in the top row of (3.5) become zero because (3.4) is a minimal resolution. It remains to show that no non-zero element of Q n b in graded degree (n b)d maps to (x b+1,..., x n )K n b 1 (g 1,..., g n ) under the Koszul differential d n b. Fix an element z = c i1,...,i n b e i1 e in b ( Q n b )(n b)d, i 1 < <i n b, i 1 b where c i1,...,i n b k, such that d n b (z) = 0 K n b 1 (g 1,..., g n )/(x b+1,..., x n )K n b 1 (g 1,..., g n ). Consider {j 1,..., j n b 1 } {b + 1,..., n}. Then the projection of d n b (z) to the e j1 e jn b 1 component of K n b 1 (g 1,..., g n )/(x b+1,..., x n )K n b 1 (g 1,..., g n ) is c j,j1,...,j n b 1 g j (x 1,..., x b, 0,..., 0)e j1 e jn b 1. j {1,...,b} Since the elements g 1 (x 1,..., x b, 0,..., 0),..., g b (x 1,..., x b, 0,..., 0) are linearly independent in k[x 1,..., x b ], we conclude that c j,j1,...,j n b 1 = 0 for all j {1,..., b} and all {j 1,..., j n b 1 } {b + 1,..., n}. Next, consider j 1 {1,..., b} and {j 2,..., j n b 1 } {b + 1,..., n}. Then the projection of d n b (z) to the e j1 e jn b 1 component is c j,j1,...,j n b 1 g j (x 1,..., x b, 0,..., 0)e j1 e jn b 1. j {1,...,b}\{j 1 } By the same reasoning as above, we conclude that c j,j1,...,j n b 1 = 0 for all {j, j 1 } {1,..., b} and {j 2,..., j n b 1 } {b + 1,..., n}. Proceeding by ascending induction

11 STABILITY OF ASSOCIATED FORMS 11 on the cardinality of {j 1,..., j n b } {1,..., b}, we see that c j1,...,j n b = 0 for all indices. We now proceed to prove that m n b = 0 implies d 1 m 1 = 0. To lighten notation, we let T be the exact complex given by the top row in the diagram (3.6), so that T i := S l i ( N i + 1) for i = 1,..., n b and T 0 := S. Claim 3.8. For i = 0,..., n b 1, there exist homomorphisms h i : T i Q i+1 and a homomorphism h 0 : S Q 1 of graded S-modules such that m i = d i+1 h i + h i 1 d i, for every i = 1,..., n b 1. Proof. To begin, note that the top row of (3.6) is the resolution of the S-module S/J. Since Ext j (S/J, S) vanishes for j < codim J = dim S dim(s/j) = n b, the dual of the top row of (3.6), that is, the complex 0 Hom(T 0, S) Hom(T 1, S) Hom(T n b, S) is exact. In particular, since Q i is a free S-module for every i 1, we see that Hom(T i 1, Q i ) Hom(T i, Q i ) Hom(T i+1, Q i ) is exact for every i 1. We now proceed to prove the claim by descending induction on i, beginning with i = n b 1. We set h n b 1 = 0. Note that m n b 1 ker ( Hom(T n b 1, Q n b 1 ) Hom(T n b, Q n b 1 ) ) by virtue of m n b = 0. It follows that there exists h n b 2 Hom(T n b 2, Q n b 1 ) such that m n b 1 = h n b 2 d n b 1 = d n b h n b 1 + h n b 2 d n b 1. This proves the claim for i = n b 1. Suppose the claim is established for i 2. Then, by the induction hypothesis, we have m i 1 d i h i 1 ker ( Hom(T i 1, Q i 1 ) Hom(T i, Q i 1 ) ). It follows that there exists h i 2 Hom(T i 2, Q i 1 ) such that This concludes the proof of the claim. m i 1 d i h i 1 = h i 2 d i 1. To finalize the proof of Lemma 3.3, it remains to observe that h 0 Hom(S, Q 1 ) must be zero because h 0 is a homomorphism of graded S-modules and Q 1 = S( d) b. It follows that m 1 = d 2 h 1 + h 0 d 1 = d 2 h 1 and so as desired. d 1 m 1 = d 1 d 2 h 1 = 0,

12 12 MAKSYM FEDORCHUK AND ALEXANDER ISAEV Lemma 3.9. Suppose g 1,..., g n is a balanced complete intersection ideal in k[x 1,..., x n ] such that (g b+1,..., g n ) (x b+1,..., x n ) and (x b+1,..., x n ) (n b)(d 1)+1 (g b+1,..., g n ). Then g b+1,..., g n k[x b+1,..., x n ]. Proof of Lemma 3.9. Since (x b+1,..., x n ) (n b)(d 1)+1 (g b+1,..., g n ), it follows that g j := g j (0,..., 0, x b+1,..., x n ), j = b + 1,..., n, form a regular sequence in k[x b+1,..., x n ]. Consider the balanced complete intersection ideals J := (x d 1,..., x d b, g b+1,..., g n ), J := (x d 1,..., x d b, g b+1,..., g n), each of which is generated by a regular sequence in S. To establish the lemma, it suffices to show that J = J, which by Lemma 2.1 follows from the equality J n(d 1) = J n(d 1). We only need to prove that J n(d 1) J n(d 1), and in fact that Fix an element (g b+1,..., g n) n(d 1) J n(d 1). n i=b+1 c i g i (g b+1,..., g n) n(d 1), where c i s are forms of degree n(d 1) d. Since x d 1,..., x d b J and k[x b+1,..., x n ] (n b)(d 1)+1 J, we can assume that c i x d 1 1 x d 1 b k[x b+1,..., x n ] (n b)(d 1) d. As we have g i g i (mod (x 1,..., x b )), it follows that n n c i g i c i g i (mod (x d 1,..., x d b)), i=b+1 i=b+1 which shows that n i=b+1 c ig i J as required. This finishes the proof of Proposition Preservation of polystability In this section, we prove Theorem 2.6, which is the main result of this paper. Take U Grass(n, S d ) Res and let I := (g g U) S. Since the field k is perfect, we can use the Hilbert-Mumford numerical criterion to analyze the GIT stability of A(U). So for any non-trivial 1-PS ρ of SL(n) we choose a basis x 1,..., x n of S 1 on which ρ acts diagonally with weights w 1 w 2 w n. Note that ρ acts with opposite weights on the dual basis z 1,..., z n of D 1. To apply the numerical criterion to the form A(U) Pk[z 1,..., z n ] n(d 1), we observe that by (2.5) a monomial z d 1 1 zn dn of

13 STABILITY OF ASSOCIATED FORMS 13 degree n(d 1) appears with a non-zero coefficient in A(U) if and only if x d 1 1 x dn n / I n(d 1). The following lemma allows us to produce such monomials. Lemma 4.1 (Grevlex Lemma). Fix 1 a n and N 0. Suppose ( (xa+1,..., x n ) N) n(d 1) I n(d 1). Let M = x d 1 ( 1 x dn n be the smallest with respect to < grevlex monomial in (xa+1,..., x n ) N) that does not lie in I n(d 1) n(d 1). Then for every i = 1,..., a we have d d i i(d 1). In particular, if M = x d 1 1 x dn n is the smallest with respect to < grevlex monomial in k[x 1,..., x n ] n(d 1) \ I n(d 1), then for every i = 1,..., n we have d d i i(d 1). Proof. We only use the fact that I is generated in degree d and has codimension n in S. By way of contradiction, suppose that d d i > i(d 1) for some i a. Choose a basis g 1,..., g n in U and let J k[x 1,..., x i ] be the ideal generated by the forms g j (x 1,..., x i, 0,..., 0), for j = 1,..., n. Then dim k[x 1,..., x i ]/J = dim S/(g 1,..., g n, x i+1,..., x n ) = 0. Hence by Lemma 2.7, (k[x 1,..., x i ]/J) K = 0 for all K > i(d 1). Thus x d 1 1 x d i i J, and so x d 1 1 x d i i I + (x i+1,..., x n ). We conclude that M = x d 1 1 x d i i xd i+1 i+1 xdn n I + (x i+1,..., x n )x d i+1 i+1 xdn n. Since M / I n(d 1), there is a monomial M ( (x i+1,..., x n )x d i+1 i+1 ) xdn n that n(d 1) does not lie in I n(d 1) either. As i a, we clearly have M (x a+1,..., x n ) N. However, M < grevlex M, which contradicts our choice of M Proof of semistability. Let M = x d 1 1 x dn n be the smallest with respect to < grevlex monomial of degree n(d 1) that does not lie in I n(d 1). Then M := z d 1 1 zn dn appears with a non-zero coefficient in A(U). By Lemma 4.1, we have Hence the ρ-weight of z d 1 1 z dn n d d i i(d 1), for all 1 i n. is (w 1 d w n d n ) 0. The Hilbert-Mumford numerical criterion implies that A(U) is semistable.

14 14 MAKSYM FEDORCHUK AND ALEXANDER ISAEV Proof of polystability: decomposable case. To prove Theorem 2.6, we proceed by induction on n. The base case is n = 1, where the statement is obvious because the only balanced complete intersection ideal for n = 1 is (x d 1) k[x 1 ] and the corresponding associated form is z1 d 1, up to a non-zero scalar. Suppose that the theorem is established for all positive integers less than a given n 2 and U Grass(n, S d ) Res is polystable. If U is decomposable, then by Corollary 2.10, we can assume that for some 1 a n 1, we have a decomposition U = U 1 U 2, where U 1 Grass(a, k[x 1,..., x a ] d ) Res and U 2 Grass(n a, k[x a+1,..., x n ] d ) Res. By Lemma 2.11, we have A(U) = A(U 1 )A(U 2 ). Since U 1 and U 2 are polystable with respect to SL(a) and SL(n a) actions, respectively, the induction hypothesis and the following standard result finalizes the proof in the case of a decomposable U: Lemma 4.2. Let V = V 1 V 2, with n i := dim V i 1. Suppose F 1 Sym d 1 V 1 and F 2 Sym d 2 V 2, where n 1 d 2 = n 2 d 1. Then F := F 1 F 2 considered as an element of Sym d 1+d 2 is SL(V )-polystable if and only if F i is SL(V i )-polystable for each i. Proof. Let λ be a one-parameter subgroup of SL(V ) such that V 1 is the weight space of λ with weight n 2 and V 2 is the weight space of λ with weight n 1. Then λ stabilizes F by the assumption n 1 d 2 = n 2 d 1. The centralizer of λ in SL(V ) is C SL(V ) (λ) = ( GL(V 1 ) GL(V 2 ) ) SL(V ). Since char(k) = 0, [16, Corollary 4.5(a)] (see also [17, Corollaire 2 and Remarque 1]) applies, and so the SL(V )-orbit of F is closed if and only if the C SL(V ) (λ)-orbit of F is ( closed. However, by the assumption n 1 d 2 = n 2 d 1, every element of the center of GL(V1 ) GL(V 2 ) ) SL(V ) acts on F as multiplication by a root of unity. It follows that the C SL(V ) (λ)-orbit of F is closed if and only if the SL(V 1 ) SL(V 2 )-orbit of F is closed, i.e., if and only if the SL(V i )-orbit of F i is closed for i = 1, Proof of polystability: indecomposable case. Suppose U Grass(n, S d ) Res is indecomposable. We will use the notation and keep in mind the conclusion of Assume that for some ρ the limit lim t 0 ρ(t) A(U) exists. This implies that w ρ (M) = 0, where w ρ (M) is the ρ-weight of M. From this we deduce a number of preliminary results. Lemma 4.3. Let 1 a n 1 be the index such that w a+1 = = w n and w a < w a+1. Then d d a = a(d 1). Proof. Assuming that d d a a(d 1) 1, we see ( n a 1 ) ( n ) 0 = w ρ (M) = d i w i > d i w i + (d a + 1)w a + d i 1 w n 0, i=1 which is impossible. i=1 i=a+1

15 STABILITY OF ASSOCIATED FORMS 15 Lemma 4.4. Let a be the integer introduced in Lemma 4.3. Then the homomorphic image of I in k[x 1,..., x n ]/(x a+1,..., x n ) k[x 1,..., x a ] is a balanced complete intersection ideal of type (d) a in k[x 1,..., x a ]. Proof. Denote the image ideal by J and suppose that J is not a complete intersection ideal. Then by Lemma 2.7 we have J a(d 1) = (x 1,..., x a ) a(d 1), and Lemma 4.3 implies x d 1 1 x da a J a(d 1). This means But then x d 1 1 x da a I a(d 1) + (x a+1,..., x n ). M = x d 1 1 x da a x d a+1 a+1 x dn n I + (x a+1,..., x n )x d a+1 a+1 x dn n. Since M I n(d 1), there is a monomial M ((x a+1,..., x n )x d a+1 a+1 x dn n ) n(d 1) that does not lie in I n(d 1) either. However, M < grevlex M, which is a contradiction. Lemma 4.5. We have k[x a+1,..., x n ] (n a)(d 1)+1 I (n a)(d 1)+1. Proof. By Lemma 2.1, it suffices to show that ( (x a+1,..., x n ) (n a)(d 1)+1) n(d 1) I n(d 1). Assume the opposite and let L = x c 1 1 x c 2 2 x ca a x c a+1 a+1 x cn n be the smallest with respect to < grevlex monomial in ( (x a+1,..., x n ) (n a)(d 1)+1) that is not in n(d 1) I n(d 1). Then by Lemma 4.1, for every i a, we have c c i i(d 1). Moreover, c c a a(d 1) 1 by assumption. But then a 1 w ρ (L) > c i w i + (c a + 1)w a + (n a)(d 1)w n 0. i=1 As L has positive ρ-weight, it must lie in I n(d 1) by the assumption that the limit lim t 0 ρ(t) A(U) exists, which contradicts our choice of L. Lemmas 4.4 and 4.5 imply that both conditions (A) and (B) of Proposition 3.1 are satisfied. Hence U is decomposable, contradicting our assumption. This proves that for every one-parameter subgroup ρ of SL(n) the limit lim t 0 ρ(t) A(U) does not exist. By the Hilbert-Mumford numerical criterion we then see that A(U) is polystable. We note that our proof in fact gives a more technical version of Theorem 2.6. Theorem 4.6. Suppose U Grass(n, S d ) Res. If U is indecomposable, then for every one-parameter subgroup ρ of SL(n) defined over k, the limit lim t 0 ρ(t) A(U) does not exist. In particular, A(U) is polystable. Furthermore, if U is indecomposable over k, then A(U) is stable.

16 16 MAKSYM FEDORCHUK AND ALEXANDER ISAEV Notice that, over a non-closed field k, the indecomposability of U Grass(n, S d ) Res does not imply on its own that A(U) is stable. Indeed, if U is indecomposable over k, it is possible for U to be decomposable over k. For example, A ( (x 1 + ix 2 ) d + (x 1 ix 2 ) d, i((x 1 + ix 2 ) d (x 1 ix 2 ) d ) ) = (z z 2 2) d 1 is not stable while the balanced complete intersection is defined and indecomposable over R. We also note that Theorem 4.6 has a curious consequence for the classification of polystable points in Grass(n, S d ) Res : Corollary 4.7. Suppose U Grass(n, Sym d V ) Res. Then U is polystable with respect to SL(V )-action if and only if there is a decomposition V = m i=1 V i, where m 1 and n i := dim k V i 1, and indecomposable U i Grass(n i, Sym d V i ) Res such that m (4.8) U = U i. i=1 Proof. Applying Corollary 2.10 repeatedly, we see that a polystable U is of the form given by Equation (4.8) with each U i indecomposable. It remains to show that every U with such decomposition is polystable. By Theorem 4.6 each A(U i ) is polystable, and since A(U) = A(U 1 ) A(U m ) by Lemma 2.11, the polystability of A(U) follows by Lemma 4.2. Since A is an SL(V )-equivariant locally closed immersion by [2, 2.5], this implies the polystability of U. 5. Invariant-theoretic variant of the Mather-Yau theorem As before, we continue to work over an arbitrary field k of characteristic 0. Fix d 2 and let (S d+1 ) be the affine open subset of smooth forms in S d+1. An element F (S d+1 ) defines an isolated homogeneous hypersurface singularity F (x 1,..., x n ) = 0. The Jacobian ideal J F := ( F/ x 1,..., F/ x n ) is a balanced complete intersection ideal, and so the Milnor algebra M F := S/J F has a Macaulay inverse system given by the associated form A(F ) := A( F/ x 1,..., F/ x n ) D n(d 1). The morphism A: (S d+1 ) D n(d 1) gives rise to an SL(n)-contravariant S d+1 D n(d 1) (see [3] and [15] for details). We will say that for F, G (S d+1 ), two singularities F = 0 and G = 0 are isomorphic if and only if (5.1) k[[x1,..., x n ]]/(F ) k[[x 1,..., x n ]]/(G) as algebras over the algebraic closure k of k. It is standard that Equation (5.1) is equivalent to GL(n) F = GL(n) G. The equality of orbits is also equivalent to

17 STABILITY OF ASSOCIATED FORMS 17 the existence of a matrix C GL(n), defined over k, such that G = C F. (Note that such a C does not have to exist over k as the example of F = x 4 1 x 4 2 and G = x x 4 2 in R[x 1, x 2 ] illustrates.) Our results imply that the morphism A sends smooth forms to polystable forms, and from this fact we deduce an invariant-theoretic version of the famous Mather- Yau theorem (see [19]). Theorem 5.2. There exists a finite collection of relative GL(n)-invariants I 1,..., I N on D n(d 1), defined over k, such that for any two smooth forms F, G S d+1, the isolated homogeneous hypersurface singularities F = 0 and G = 0 are isomorphic if and only if [I 1 (A(F )) : : I N (A(F ))] = [I 1 (A(G)) : : I N (A(G))]. Remark 5.3. Our results show that the Mather-Yau theorem in the homogeneous case can be extended to the case of an arbitrary field k of characteristic 0 by stating that for F, G (S d+1 ), the singularities F = 0 and G = 0 are isomorphic if and only if M F k k and MG k k are isomorphic as k-algebras. The main novelty of Theorem 5.2 is in showing that one can check whether such an isomorphism exists simply by evaluating finitely many SL(n)-invariants on the associated forms of M F and M G, and that this can be done without passing to the algebraic closure of k. To prove this result, we will need the following immediate consequence of Theorem 2.6 for the geometry of the associated form morphism (cf. 2.3): A: Grass(n, S d ) Res PD n(d 1). Corollary 5.4. The induced morphism of GIT quotients A// SL(n): Grass(n, S d ) Res // SL(n) PD ss n(d 1)// SL(n) is a locally closed immersion. Proof. By [2, 2.5], Grass(n, S d ) Res maps isomorphically via A to an SL(n)-invariant open subset, say O, of an SL(n)-invariant closed subscheme Z PD n(d 1). By [12, Theorem 1.2], the image of A lies in the semistable locus Z ss, and Theorem 2.6 implies that O is a saturated open subset of Z ss. The corollary now follows. Proof of Theorem 5.2. Note that the two singularities are isomorphic if and only if GL(n) F = GL(n) G in S d+1, which by the GIT stability of smooth forms is equivalent to the fact that F and G map to the same point in the GIT quotient P(S d+1 ) // SL(n). Recall from [2, 2.3] that the (projectivized) morphism A: P(S d+1 ) PD n(d 1) factors as the composition of the gradient morphism : P(S d+1 ) Grass(n, S d ) Res, defined by F := F/ x 1,..., F/ x n, and the associated form morphism A: Grass(n, S d ) Res PD n(d 1).

18 18 MAKSYM FEDORCHUK AND ALEXANDER ISAEV By Corollary 5.4, the induced morphism of GIT quotients A// SL(n): Grass(n, S d ) Res // SL(n) PD ss n(d 1)// SL(n) is a locally closed immersion. Next, by [12, Theorem 1.1], we have that (F ) is polystable for every F P(S d+1 ). Moreover, by [12, Proposition 2.1 (2)] the induced morphism on the GIT quotients // SL(n): P(S d+1 ) // SL(n) Grass(n, S d ) Res // SL(n) is injective. We conclude that the induced morphism A// SL(n): P(S d+1 ) // SL(n) PD ss n(d 1)// SL(n) is injective. The theorem now follows from the definition of the GIT quotient PDn(d 1) ss // SL(n) and the fact that the ring of SL(n)-invariant forms on D n(d 1) is finitely generated by the Gordan-Hilbert theorem. Example 5.5. We conclude with an example showing that our GIT stability results are optimal as far as complete intersection Artinian algebras are concerned. Consider the following quasi-homogeneous form: Then the Milnor algebra of F is F (x 1,..., x n ) = x d x dn+1 n. M F = S/(x d 1 1,..., x dn n ), which is a complete intersection Artin k-algebra of socle degree (d d n ) n. The homogeneous Macaulay inverse system of this algebra (up to a non-zero scalar) is z d zn dn 1. Unless d 1 = = d n, this form is patently unstable with respect to the SL(n)-action on D (d1 + +d n) n, and so all relative GL(n)-invariants vanish on it. References [1] Jarod Alper and Alexander Isaev. Associated forms in classical invariant theory and their applications to hypersurface singularities. Math. Ann., 360(3-4): , [2] Jarod Alper and Alexander Isaev. Associated forms and hypersurface singularities: The binary case. J. reine angew. Math., To appear, DOI: /crelle [3] Jarod Alper, Alexander Isaev, and Nikolay Kruzhilin. Associated forms of binary quartics and ternary cubic. Transform. Groups, 21(3): , [4] David Bayer and Michael Stillman. A criterion for detecting m-regularity. Invent. Math., 87(1):1 11, [5] Max Benson. Analytic equivalence of isolated hypersurface singularities defined by homogeneous polynomials. In Singularities (Arcata, CA, 1981), volume 40 of Proc. Sympos. Pure Math., pages Amer. Math. Soc., Providence, RI, [6] David A. Buchsbaum and David Eisenbud. Generic free resolutions and a family of generically perfect ideals. Advances in Math., 18(3): , 1975.

19 STABILITY OF ASSOCIATED FORMS 19 [7] David A. Buchsbaum and Dock S. Rim. A generalized Koszul complex. II. Depth and multiplicity. Trans. Amer. Math. Soc., 111: , [8] Michael Eastwood and Alexander Isaev. Extracting invariants of isolated hypersurface singularities from their moduli algebras. Math. Ann., 356(1):73 98, [9] Michael Eastwood and Alexander Isaev. Invariants of Artinian Gorenstein algebras and isolated hypersurface singularities. In Developments and retrospectives in Lie theory, volume 38 of Dev. Math., pages Springer, Cham, [10] David Eisenbud. Commutative algebra with a view toward algebraic geometry, volume 150 of Graduate Texts in Mathematics. Springer-Verlag, New York, [11] Maksym Fedorchuk. Direct sum decomposability of smooth polynomials, Preprint, arxiv: [12] Maksym Fedorchuk. GIT semistability of Hilbert points of Milnor algebras. Math. Ann., 367(1-2): , [13] Anthony Iarrobino and Vassil Kanev. Power sums, Gorenstein algebras, and determinantal loci, volume 1721 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, [14] Alexander Isaev. A criterion for isomorphism of Artinian Gorenstein algebras. J. Commut. Algebra, 8(1):89 111, [15] Alexander Isaev. On the contravariant of homogeneous forms arising from isolated hypersurface singularities. Internat. J. Math., 27(12): , [16] George R. Kempf. Instability in invariant theory. Ann. of Math. (2), 108(2): , [17] Domingo Luna. Adhérences d orbite et invariants. Invent. Math., 29(3): , [18] Francis S. Macaulay. The algebraic theory of modular systems. Cambridge Mathematical Library. Cambridge University Press, Cambridge, Revised reprint of the 1916 original. [19] John Mather and Stephen S.-T. Yau. Classification of isolated hypersurface singularities by their moduli algebras. Invent. Math., 69(2): , [20] David Mumford, John Fogarty, and Frances Kirwan. Geometric invariant theory, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Springer-Verlag, Berlin, third edition, [21] Günter Scheja and Uwe Storch. Über Spurfunktionen bei vollständigen Durchschnitten. J. reine angew. Math., 278/279: , (Fedorchuk) Department of Mathematics, Boston College, 140 Commonwealth Ave, Chestnut Hill, MA 02467, USA address: maksym.fedorchuk@bc.edu (Isaev) Mathematical Sciences Institute, Australian National University, Acton, ACT 2601, Australia address: alexander.isaev@anu.edu.au