Irreducibility of the Gorenstein loci of Hilbert schemes via ray families
aa r X i v : . [ m a t h . AG ] N ov Irreducibility of the Gorenstein loci of Hilbert schemes via rayfamilies
Gianfranco Casnati, Joachim Jelisiejew, Roberto Notari ∗ September 18, 2018
Abstract
We analyse the Gorenstein locus of the Hilbert scheme of d points on P n i.e. the opensubscheme parameterising zero-dimensional Gorenstein subschemes of P n of degree d . Wegive new sufficient criteria for smoothability and smoothness of points of the Gorenstein locus.In particular we prove that this locus is irreducible when d ≤ and find its componentswhen d = 14 .The proof is relatively self-contained and it does not rely on a computer algebra sys-tem. As a by–product, we give equations of the fourth secant variety to the d -th Veronesereembedding of P n for d ≥ . keywords: Hilbert scheme of points, smoothability, Gorenstein algebra, secant variety.
MSC classes:
Let k be an algebraically closed field of characteristic neither nor and denote by H ilb p ( t ) P N theHilbert scheme parameterising closed subschemes in P N with fixed Hilbert polynomial p ( t ) ∈ Q [ t ] .Since A. Grothendieck proved the existence of such a parameter space in 1966 (see [Gro95]), theproblem of dealing with H ilb p ( t ) P N and its subloci has been a fruitful field attracting the interestof many researchers in algebraic geometry.Only to quickly mention some of the classical results which deserve, in our opinion, a par-ticular attention, we recall Hartshorne’s proof of the connectedness of H ilb p ( t ) P N (see [Har66]),the description of the locus of codimension arithmetically Cohen–Macaulay subschemes dueto G. Ellingsrud and J. Fogarty (see [Fog68] for the dimension zero case and [Ell75] for largerdimension) and of the study of the locus of codimension arithmetically Gorenstein subschemesdue to J. Kleppe and R.M. Miró–Roig (see [MR92] and [KMR98]).If we restrict our attention to the case of zero–dimensional subschemes of degree d , i.e. sub-schemes with Hilbert polynomial p ( t ) = d , then the first significant results are due to J. Fogarty(see [Fog68]) and to A. Iarrobino (see [Iar72]).In [Fog68], the author proves that H ilb d P is smooth, hence irreducible thanks to Hartshorne’sconnectedness result (the same result holds, when one substitutes P by any smooth surface). ∗ The first and third authors are supported by the framework of PRIN 2010/11 “Geometria delle varietà al-gebriche”, cofinanced by MIUR, and are members of GNSAGA of INdAM. The second author was partiallysupported by the project “Secant varieties, computational complexity, and toric degenerations” realised withinthe Homing Plus programme of Foundation for Polish Science, co-financed from European Union, Regional De-velopment Fund. The second author is a doctoral fellow at the Warsaw Center of Mathematics and ComputerScience financed by the Polish program KNOW. This paper is a part of “Computational complexity, generalisedWaring type problems and tensor decompositions” project within “Canaletto”, the executive program for scientificand technological cooperation between Italy and Poland, 2013–2015. This article is partially supported by PolishNational Science Center, project 2014/13/N/ST1/02640.
1n the other hand in [Iar72], A. Iarrobino deals with the reducibility when N ≥ and d islarge with respect to N . In order to better understand the result, recall that the locus of reducedschemes R ⊆ H ilb d P N is birational to a suitable open subset of the d -th symmetric product of P N , thus it is irreducible of dimension dN . We will denote by H ilb gend P N its closure in H ilb d P N .It is a well–known and easy fact that H ilb gend P N is an irreducible component of dimension dN , byconstruction. In [Iar72], the author proves that H ilb d P N is never irreducible when d ≫ N ≥ ,showing that there is a family of schemes of dimension greater than dN . Such a family is thusnecessarily contained in a component different from H ilb gend P N .D.A. Cartwright, D. Erman, M. Velasco, B. Viray proved that already for d = 8 and N ≥ ,the scheme H ilb d P N is reducible (see [CEVV09]).In view of these earlier works it seems reasonable to consider the irreducibility and smoothnessof open loci in H ilb d P N defined by particular algebraic and geometric properties. In the presentpaper we are interested in the locus H ilb Gd P N of points in H ilb d P N representing schemes whichare Gorenstein. This is an important locus: e.g. it has an irreducible component H ilb G,gend P N := H ilb gend P N ∩ H ilb Gd P N of dimension dN containing all the points representing reduced schemes.Moreover it is open, but in general not dense, inside H ilb d P N . Recently, interesting interactionsbetween H ilb Gd P N and the geometry of secant varieties and general topology have been found(see for example [BB14], [BJJM]).Some results about H ilb Gd P N are known. The irreducibility and smoothness of H ilb Gd P N when N ≤ is part of the folklore (see [CN09, Cor 2.6] for more precise references). When N ≥ , theproperties of H ilb Gd P N have been object of an intensive study in recent years.E.g., it is classically known that H ilb Gd P N is never irreducible for d ≥ and N ≥ , at leastwhen the characteristic of k is zero (see [IE78] and [IK99]: see also [CN11]). Also for N = 4 and d ≥ or N = 5 and d ≥ the scheme H ilb Gd P N is reducible, see [BB14, Section 6, p. 81]. Forfixed N ∈ { , } the minimal value of d , for which this scheme is reducible, is not known.As reflected by the quoted papers, it is natural to ask if H ilb Gd P N is irreducible when d ≤ and N is arbitrary. There is some evidence of an affirmative answer to this question. Indeedthe first and third authors studied the locus H ilb Gd P N when d ≤ and N is arbitrary, provingits irreducibility and dealing in detail with its singular locus in a series of papers [CN09, CN11,CN14, CN13].A key point in the study of a zero–dimensional scheme X ⊆ P N is that it is abstractlyisomorphic to Spec A where A is an Artin k -algebra with dim k ( A ) = d . Moreover the irreduciblecomponents of such an X correspond bijectively to those direct summands of A , which are local.Thus, in order to deal with H ilb d P N , it suffices to deal with the irreducible schemes in H ilb d ′ P N for each d ′ ≤ d .In all of the aforementioned papers, the methods used in the study of H ilb Gd P N rely on analmost explicit classification of the possible structure of local, Artin, Gorenstein k -algebras oflength d . Once such a classification is obtained, the authors prove that all the corresponding irre-ducible schemes are smoothable, i.e. actually lie in H ilb G,gend P N . To this purpose they explicitlyconstruct a projective family flatly deforming the scheme they are interested in (or, equivalently,the underlying algebra) to reducible schemes that they know to be in H ilb G,gend P N because theircomponents have lower degree.Though such an approach sometimes seems to be too heavy in terms of calculations, onlythanks to such a partial classification it is possible to state precise results about the singularitiesof H ilb Gd P N .However, in the papers [CN11, CN14], there are families H d of schemes of degree d , where d = 10 , , for which an explicit algebraic description in the above sense cannot be obtained (seeSection 3 of [CN11] for the case d = 10 , Section 4 of [CN14] for d = 11 ). Nevertheless, using analternative approach the authors are still able to prove the irreducibility of H ilb Gd P N and studyits singular locus. Indeed, using Macaulay’s theory of inverse systems, the authors check the2rreducibility of the aforementioned loci H d inside H ilb Gd P N . Then they show the existence of asmooth point in H d ∩ H ilb G,gend P N . Hence, it follows that H d ⊆ H ilb G,gend P N .The aim of the present paper is to refine and generalise this method. First, we avoid a caseby case approach by analysing large classes of algebras. Second, in [CN11, CN14] a direct check(e.g. using a computer algebra program) is required to compute the dimension of tangent spaceto the Hilbert scheme at some specific points to conclude that they are smooth. We avoid theneed of such computations by exhibiting classes of points which are smooth, making the paperself–contained.Using this method, we finally prove the following two statements. Theorem A.
If the characteristic of k is neither nor , then H ilb Gd P N is irreducible of dimen-sion dN for each d ≤ and for d = 14 and N ≤ . Theorem B.
If the characteristic of k is and N ≥ , then H ilb G P N is connected and it hasexactly two irreducible components, which are generically smooth. Theorem A has an interesting consequence regarding secant varieties of Veronese embeddings.In [Ger99] Geramita conjectures that the ideal of the nd secant variety (the variety of secantlines) of the d th Veronese embedding of P n is generated by the × minors of the i th catalecticantmatrix for ≤ i ≤ d − . Such a conjecture was confirmed in [Rai12]. As pointed out in [BB14,Section 8.1], the above Theorem A allows to extend the above result as follows: if r ≤ , r ≤ d and, then for every r ≤ i ≤ d − r the set–theoretic equations of the r th secant variety of the d th Veronese embedding of P n are given by the ( r + 1) × ( r + 1) minors of the i th catalecticantmatrix.The proofs of Theorem A and Theorem B are highly interlaced and they follow from a longseries of partial results. In order to better explain the ideas and methods behind their proofs wewill describe in the following lines the structure of the paper.In our analysis we incorporate several tools. In Section 2 we recall the classical ones, most no-tably Macaulay’s correspondence for local, Artinian, Gorenstein algebras and Macaulay’s GrowthTheorem. Moreover we also list some criteria for checking the flatness of a family of algebraswhich will be repeatedly used throughout the whole paper.In Section 3 we analyse Artin Gorenstein quotients of a power series ring and exploit the richautomorphism group of this ring to put the quotient into suitable standard form, deepening aresult by A. Iarrobino.In Section 4 we further analyse the quotients, especially their dual socle generators. We alsoconstruct several irreducible subloci of the Hilbert scheme using the theory of secant varieties.We give a small contribution to this theory, showing that the fourth secant variety to a Veronesereembedding of P n is defined by minors of a suitable catalecticant matrix.Section 5 introduces a central object in our study: a class of families, called ray families, forwhich we have relatively good control of the flatness and, in special cases, fibers. Most notably,Subsection 5.2 gives a class of tangent preserving flat families, which enable us to constructsmooth points on the Hilbert scheme of points without the necessity of heavy computations.Finally, in Section 6, we give the proofs of Theorem A and B. It is worth mentioning thatthese results are rather easy consequences of the introduced machinery. In this section we alsoprove the following general smoothability result (see Thm 6.14), which has no restriction on thelength of the algebra and generalises the smoothability results from [Sal79], [CN13] and [EV11]. Theorem C.
Let k be an algebraically closed field of characteristic neither nor . Let A be alocal Artin Gorenstein k -algebra with maximal ideal m .If dim k ( m / m ) ≤ and dim k ( m / m ) ≤ , then Spec A is smoothable. otation All symbols appearing below are defined in Section 2. k an algebraically closed field of characteristic = 2 , . P = k [ x , . . . , x n ] a polynomial ring in n variables and fixed basis. S = k [[ α , . . . , α n ]] a power series ring dual (see Subsection 2.2) to P , with a fixed (dual)basis. m S the maximal ideal of S . S poly = k [ α , . . . , α n ] a polynomial subring of S defined by the choice of the basis. H A the Hilbert function of a local Artin algebra A . ∆ A,i , ∆ i the i -th row of the symmetric decomposition of the Hilbert function ofa local Artin Gorenstein algebra A as in Theorem 2.3. e ( a ) the a -th “embedding dimension”, equal to P at =0 ∆ t (1) , as in Defini-tion 3.1. ann S ( f ) the annihilator of f ∈ P with respect to the action of S . Apolar ( f ) the apolar algebra of f ∈ P , equal to S/ ann S ( f ) . Let n be a natural number. By ( S, m S , k ) we denote the power series ring k [[ α , . . . , α n ]] ofdimension n with a fixed basis α , . . . , α n . The chosen basis determines a polynomial ring S poly = k [ α , . . . , α n ] ⊆ S . By P we denote the polynomial ring k [ x , . . . , x n ] . We will laterdefine a duality between S and P , see Subsection 2.2. We usually think of n being large enough,so that the considered local Artin algebras are quotients of S .For an element f ∈ P , we say that f does not contain x i if f ∈ k [ x , . . . , x i − , x i +1 , . . . , x n ] ;similarly for σ ∈ S or σ ∈ S poly . For f ∈ P , by f d we denote the degree d part of f , with respectto the total degree; similarly for σ ∈ S .By P m and P ≤ m we denote the space of homogeneous polynomials of degree m and (not nec-essarily homogeneous) polynomials of degree at most m respectively. These spaces are naturallyaffine spaces over k , which equips them with a scheme structure.Recall that S has a rich automorphism group: for every choice of elements σ , . . . , σ n ∈ m S linearly independent in m S / m S there is a unique automorphism ϕ of S such that ϕ ( α i ) = σ i for i = 1 , , . . . , n . The existence of such automorphisms is employed in Section 4 to put theconsidered Artin Gorenstein algebras in a better form. See e.g. [ER15, Section 2] for details andexamples of this method. Remark 2.1.
For the reader’s convenience we introduce numerous examples, which illustratethe possible applications. In all these examples k may have arbitrary characteristic = 2 , unlessotherwise stated. In this section we recall the basic facts about Artin Gorenstein algebras. For a more thoroughtreatment we refer to [IK99], [Eis95], [CN09] and [Jel13].Finite type zero-dimensional schemes correspond to Artin algebras. Every such algebra A splits as a finite product of its localisations at maximal ideals, which corresponds to the fact thatthe support of Spec A is finite and totally disconnected. Therefore, we will focus our interest on local Artin k -algebras. Since k is algebraically closed, such algebras have residue field k .An important invariant of a local algebra ( A, m , k ) is its Hilbert function H A defined by H A ( l ) = dim k m l / m l +1 . Since H A ( l ) = 0 for l ≫ it is usual to write H A as the vector of its4on-zero values. The socle degree of A is the largest l such that H A ( l ) = 0 . Such an algebra is Gorenstein if the annihilator of m is a one-dimensional vector space over k , see [Eis95, Chap 21].We recall for reader’s benefit that a finite not necessarily local algebra A is Gorenstein if andonly if all its localisations at maximal ideals are Gorenstein (in particular it is meaningful todiscuss the irreducibility of the Gorenstein locus in the Hilbert scheme by reducing to the studyof deformations of local Gorenstein algebras: see Section 2.4).Since k is algebraically closed, we may write each Artin local algebra ( A, m , k ) as a quotient ofthe power series ring S = k [[ α , . . . , α n ]] when n is large enough, in fact n ≥ H A (1) is sufficient.Since dim k A is finite, such a presentation gives a presentation A = S poly /I , i.e. a point [Spec A ] of the Hilbert scheme of A n = Spec S poly . In this section we introduce the contraction mapping, which is closely related to Macaulay’sinverse systems. We refer to [Iar94] and [Eis95, Chap 21] for details and proofs.Recall that P = k [ x , . . . , x n ] is a polynomial ring and S = k [[ α , . . . , α n ]] is a power seriesring. The k -algebra S acts on P by contraction (see [IK99, Def 1.1]). This action is denoted by ( · ) y ( · ) : S × P → P and defined as follows. Let x a = x a . . . x a n n ∈ P and α b = α b . . . α b n n ∈ S be monomials. We write a ≥ b if and only if a i ≥ b i for all ≤ i ≤ n . Then α b y x a := ( x a − b if a ≥ b otherwise.This action extends to S × P → P by k -linearity on P and countable k -linearity on S .The contraction action induces a perfect pairing between S/ m s +1 S and P ≤ s , which restricts toa perfect pairing between the degree s polynomials in S poly and P . These pairings are compatiblefor different choices of s .If f ∈ P then a derivative of f is an element of the S -module Sf , i.e. an element of the form ∂ y f for ∂ ∈ S . By definition, these elements form an S -submodule of P , in particular a k -linearsubspace.Let A = S/I be an Artin quotient of S , then A is local. The contraction action associates to A an S -submodule M ⊆ P consisting of elements annihilated by I , so that A and M are dual. If A is Gorenstein, then the S -module M is cyclic, generated by a polynomial f of degree s equal tothe socle degree of A . We call every such f a dual socle generator of the Artin Gorenstein algebra A . Unlike M , the polynomial f is not determined uniquely by the choice of the presentation A = S/I , however if f and g are two dual socle generators, then g = ∂ y f , where ∂ ∈ S isinvertible.Conversely, let f ∈ P be a polynomial of degree s . We can associate it the ideal I := ann S ( f ) such that A := S/I is a local Artin Gorenstein algebra of socle degree s . We call I the apolarideal of f and A the apolar algebra of f , which we denote as A = Apolar ( f ) . From the discussion above it follows that every local Artin Gorenstein algebra is an apolar algebraof some polynomial.
Remark 2.2.
Recall that we may think of S/ m s +1 S as the linear space dual to P ≤ s . An automor-phism ψ of S or S/ m s +1 S induces an automorphism ψ ∗ of the k -linear space P ≤ s . If f ∈ P ≤ s and I is the apolar ideal of f , then ψ ( I ) is the apolar ideal of ψ ∗ ( f ) . Moreover, f and ψ ∗ ( f ) havethe same degree. .3 Iarrobino’s symmetric decomposition of Hilbert function One of the most important invariants possessed by a local Artin Gorenstein algebra is the sym-metric decomposition of its Hilbert function, due to Iarrobino [Iar94]. To state the theorem itis convenient to define addition of vectors of different lengths position-wise: if a = ( a , . . . , a n ) and b = ( b , . . . , b m ) are vectors, then a + b = ( a + b , . . . , a max( m,n ) + b max( m,n ) ) , where a i = 0 for i > n and b i = 0 for i > m . In the following, all vectors are indexed starting from zero.Let ( A, m , k ) be a local Artin Gorenstein algebra. By (0 : m l ) we denote the annihilator of m l in A . The chain m ) ⊆ (0 : m ) ⊆ . . . defines a filtration on A . In general, it is differentfrom the usual filtration m s +1 ⊆ m s ⊆ m s − ⊆ . . . . The analysis of mutual position of thesefiltrations is the content of Theorem 2.3 below. Theorem 2.3 (Iarrobino’s symmetric decomposition of Hilbert function) . Let ( A, m , k ) be a localArtin Gorenstein algebra of socle degree s and Hilbert function H A . Let ∆ i ( t ) := dim k (0 : m s +1 − i − t ) ∩ m t (0 : m s − i − t ) ∩ m t + (0 : m s +1 − i − t ) ∩ m t +1 for t = 0 , , . . . , s − i. The vectors ∆ , ∆ , . . . , ∆ s have the following properties:1. the vector ∆ i has length s + 1 − i and satisfies ∆ i ( t ) = ∆ i ( s − i − t ) for all integers t ∈ [0 , s − i ] .2. the Hilbert function H A is equal to the sum P si =0 ∆ i .3. the vector ∆ is equal to the Hilbert function of a local Artin Gorenstein graded algebra ofsocle degree s . Let ( A, m , k ) be a local Artin Gorenstein algebra. There are a few important remarks to do.1. Since ∆ is the Hilbert function of an algebra, we have ∆ (0) = 1 = H A (0) . Thus forevery i > we have ∆ i (0) = 0 . From symmetry it follows that ∆ i ( s + 1 − i ) = 0 . Inparticular ∆ s = (0) and ∆ s − = (0 , , so we may ignore these vectors. On the other hand ∆ s − = (0 , q, is in general non-zero and its importance is illustrated by Proposition 4.5.2. Suppose that H A = (1 , n, , for some n > . Then we have ∆ = (1 , ∗ , ∗ , and ∆ = (0 , ∗ , , thus ∆ = (1 , ∗ , , , so that ∆ = (1 , , , because of its symmetry.Then ∆ = (0 , n − , . Similarly, if H A = (1 , n, e, is the Hilbert function of a localArtin Gorenstein algebra, then n ≥ e . This is a basic example on how Theorem 2.3 imposesrestrictions on the Hilbert function of A .3. If A is graded, then ∆ = H A and all other ∆ • are zero vectors, see [Iar94, Prop 1.7].4. For every a ≤ s the partial sum P ai =0 ∆ i is the Hilbert function of a local Artin gradedalgebra, see [Iar94, Def 1.3, Thm 1.5], see also [Iar94, Subsection 1.F]. In particular itsatisfies Macaulay’s Growth Theorem, see Subsection 2.5. Thus e.g. there is no local ArtinGorenstein algebra with Hilbert function decomposition satisfying ∆ = (1 , , , , , and ∆ = (0 , , , , , because then (∆ + ∆ )(1) = 1 and (∆ + ∆ )(2) = 2 .Let us now analyse the case when A = Apolar ( f ) = S/ ann S ( f ) is the apolar algebra of apolynomial f ∈ P , where f = P si =0 f i for some f i ∈ P i . Each local Artin Gorenstein algebrais isomorphic to such algebra, see Subsection 2.2. For the proofs of the following remarks,see [Iar94].1. Vector ∆ is equal to the Hilbert function of Apolar ( f s ) , the apolar algebra of the leadingform of f . 6. If A is graded, then ann S ( f ) = ann S ( f s ) , so that we may always assume that f = f s .Moreover, in this case H A ( m ) is equal to ( Sf s ) m , the number of degree m derivatives of f s .3. Let f , f be polynomials of degree s such that f − f is a polynomial of degree d < s . Let A i = Apolar ( f i ) and let ∆ A i ,m be the symmetric decomposition of the Hilbert function H A i of A i for i = 1 , . Then ∆ A ,m = ∆ A ,m for all m < s − d , see [Iar94, Lem 1.10]. An Artin algebra A is called smoothable if it is a (finite flat) limit of smooth algebras, i.e. ifthere exists a finite flat family over an irreducible base with a special fiber isomorphic to Spec A and general fiber smooth. Recall that A ≃ A m × . . . A m r , where m i are maximal ideals of A .The algebra A is smoothable if all localisations A m at its maximal ideals are smoothable. Theconverse also holds, i.e. if an algebra A ≃ B × B is smoothable, then the algebras B and B are also smoothable, a complete and characteristic free proof of this fact will appear shortlyin [BJ]. We say that a zero-dimensional scheme Z = Spec A is smoothable if the algebra A issmoothable.It is crucial that every local Artin Gorenstein algebra A with H A (1) ≤ is smoothable,see [CN09, Prop 2.5], which follows from the Buchsbaum-Eisenbud classification of resolutions,see [BE77]. Also complete intersections are smoothable. A complete intersection Z ⊆ P n issmoothable by Bertini’s Theorem (see [Har10, Example 29.0.1], but note that Hartshorne usesa slightly weaker definition of smoothability, without finiteness assumption). If Z = Spec A is acomplete intersection in A n , then Z is a union of connected components of a complete intersection Z ′ = Spec B in P n , so that B ≃ A × C for some algebra C . The algebra B is smoothable since Z ′ is. Thus also the algebra A is smoothable, i.e. Z is smoothable. Definition 2.4.
A smoothable Artin algebra A of length d , corresponding to Spec A ⊆ P n , is unobstructed if the tangent space to H ilb d ( P n ) at the k -point [Spec A ] =: p has dimension nd .If A is unobstructed, then p is a smooth point of the Hilbert scheme. The unobstructedness is independent of n and the chosen embedding of Spec A into P n , seediscussion before [CN09, Lem 2.3]. The argument above shows that algebras corresponding tocomplete intersections in A n and P n are unobstructed. Every local Artin Gorenstein algebra A with H A (1) ≤ is unobstructed, see [CN09, Prop 2.5]. Moreover, every local Artin Gorensteinalgebra A with H A (1) ≤ is a complete intersection in A by the Hilbert-Burch theorem. Definition 2.5.
An Artin algebra A is limit-reducible if there exists a flat family (over anirreducible base) whose special fiber is A and general fiber is reducible. An Artin algebra A is strongly non-smoothable if it is not limit-reducible. Clearly, strongly non-smoothable algebras (other than A = k ) are non-smoothable. Thedefinition of strong non-smoothability is useful, because to show that there is no non-smoothablealgebra of length less than d it is enough to show that there is no strongly non-smoothablealgebra of length less than d . We will recall Macaulay’s Growth Theorem and Gotzmann’s Persistence Theorem, which providestrong restrictions on the possible Hilbert functions of graded algebras. Fix n ≥ . Let m beany natural number, then m may be uniquely written in the form m = (cid:18) m n n (cid:19) + (cid:18) m n − n − (cid:19) + . . . + (cid:18) m (cid:19) , m n > m n − > . . . > m . We define m h i i := (cid:18) m n + 1 n + 1 (cid:19) + (cid:18) m n − + 1 n (cid:19) + . . . + (cid:18) m + 12 (cid:19) . It is useful to compute some initial values of the above defined function, i.e. h n i = 1 for all n , h i = 4 , h i = 5 , h i = 10 or h i = 5 . Theorem 2.6 (Macaulay’s Growth Theorem) . If A is a graded quotient of a polynomial ringover k , then the Hilbert function H A of A satisfies H A ( m + 1) ≤ H A ( m ) h m i for all m .Proof. See [BH93, Thm 4.2.10].Note that the assumptions of Theorem 2.6 are satisfied for every local Artin k -algebra ( A, m , k ) , since its Hilbert function is by definition equal to the Hilbert function of the asso-ciated graded algebra. Remark 2.7.
We will frequently use the following easy consequence of Theorem 2.6.Let A be a graded quotient of a polynomial ring over k . Suppose that H A ( l ) ≤ l for some l .Then H A ( l ) = (cid:0) ll (cid:1) + (cid:0) l − l − (cid:1) + . . . and H A ( l ) h l i = (cid:0) l +1 l +1 (cid:1) + (cid:0) ll (cid:1) + . . . = H A ( l ) , thus H A ( l + 1) ≤ H A ( l ) .It follows that the Hilbert function of H A satisfies H A ( l ) ≥ H A ( l + 1) ≥ H A ( l + 2) ≥ . . . . Inparticular H A ( m ) ≤ l for all m ≥ l . Theorem 2.8 (Gotzmann’s persistence Theorem) . Let A = S poly /I be a graded quotient of apolynomial ring S poly over k and suppose that for some l we have H A ( l + 1) = H A ( l ) h l i and I isgenerated by elements of degree at most l . Then H A ( m + 1) = H A ( m ) h m i for all m ≥ l .Proof. See [BH93, Thm 4.3.3].In the following we will mostly use the following consequence of Theorem 2.8, for which weintroduce some (non-standard) notation. Let I ⊆ S poly = k [ α , . . . , α n ] be a graded ideal in apolynomial ring and m ≥ . We say that I is m -saturated if for all l ≤ m and σ ∈ ( S poly ) l thecondition σ · ( α , . . . , α n ) m − l ⊆ I implies σ ∈ I . Lemma 2.9.
Let S poly = k [ α , . . . , α n ] be a polynomial ring with maximal ideal n = ( α , . . . , α n ) .Let I ⊆ S poly be a graded ideal and A = S poly /I . Suppose that I is m -saturated for some m ≥ .Then1. if H A ( m ) = m + 1 and H A ( m + 1) = m + 2 , then H A ( l ) = l + 1 for all l ≤ m , in particular H A (1) = 2 .2. if H A ( m ) = m + 2 and H A ( m + 1) = m + 3 , then H A ( l ) = l + 2 for all l ≤ m , in particular H A (1) = 3 .Proof.
1. First, if H A ( l ) ≤ l for some l < m , then by Macaulay’s Growth Theorem H A ( m ) ≤ l < m + 1 , a contradiction. So it suffices to prove that H A ( l ) ≤ l + 1 for all l < m .Let J be the ideal generated by elements of degree at most m in I . We will prove that thegraded ideal J of S poly defines a P linearly embedded into P n − .Let B = S poly /J . Then H B ( m ) = m + 1 and H B ( m + 1) ≥ m + 2 . Since H B ( m ) = m + 1 = (cid:0) m +1 m (cid:1) , we have H B ( m ) h m i = (cid:0) m +2 m +1 (cid:1) = m + 2 and by Theorem 2.6 we get H B ( m + 1) ≤ m + 2 ,thus H B ( m + 1) = m + 2 . Then by Gotzmann’s Persistence Theorem H B ( l ) = l + 1 for all l > m .This implies that the Hilbert polynomial of Proj B ⊆ P n − is h B ( t ) = t + 1 , so that Proj B ⊆ P n − is a linearly embedded P . In particular the Hilbert function and Hilbert polynomial of Proj B are equal for all arguments. By assumption, we have J l = J satl for all l < m . Then H A ( l ) = H S poly /J ( l ) = H S poly /J sat ( l ) = l + 1 for all l < m and the claim of the lemma follows.8. The proof is similar to the above one; we mention only the points, where it changes.Let J be the ideal generated by elements of degree at most m in I and B = S poly /J . Then H B ( m ) = m + 2 = (cid:0) m +1 m (cid:1) + (cid:0) m − m − (cid:1) , thus H B ( m + 1) ≤ (cid:0) m +2 m +1 (cid:1) + (cid:0) mm (cid:1) = m + 3 and B defines aclosed subscheme of P n − with Hilbert polynomial h B ( t ) = t + 2 . There are two isomorphismtypes of such subschemes: P union a point and P with an embedded double point. One checksthat for these schemes the Hilbert polynomial is equal to the Hilbert function for all argumentsand then proceeds as in the proof of Point 1. Remark 2.10. If A = S poly /I is a graded Artin Gorenstein algebra of socle degree s , then it is m -saturated for every m ≤ s . Indeed, we may assume that A = Apolar ( F ) for some homogeneous F ∈ P of degree s , then I = ann S ( F ) . Let n = ( α , . . . , α n ) ⊆ k [ α , . . . , α n ] = S poly . Take σ ∈ ( S poly ) l , then σ ∈ I if and only if σ y F = 0 . Similarly, σ n m − l ⊆ I if and only if every elementof n m − l annihilates σ y F . Since σ y F is either a homogeneous polynomial of degree s − l ≥ m − l or it is zero, both conditions are equivalent. Remark 2.11.
Clearly, if two graded ideals I and J of S poly agree up to degree m and I is m -saturated, then also J is m -saturated. Spec k [ t ] For further reference we explicitly state a purely elementary flatness criterion. Its formulation isa bit complicated, but this is precisely the form which is needed for the proofs. This criterionrelies on the easy observation that the torsion-free modules over k [ t ] are flat. Proposition 2.12.
Suppose S is a k -module and I ⊆ S [ t ] is a k [ t ] -submodule. Let I := I ∩ S .If for every λ ∈ k we have ( t − λ ) ∩ I ⊆ ( t − λ ) I + I [ t ] , then S [ t ] /I is a flat k [ t ] -module.Proof. The ring k [ t ] is a principal ideal domain, thus a k [ t ] -module is flat if and only if it istorsion-free, see [Eis95, Cor 6.3]. Since every polynomial in k [ t ] decomposes into linear factors,to prove that M = S [ t ] /I is torsion-free it is enough to show that t − λ are non–zerodivisors on M , i.e. that ( t − λ ) x ∈ I implies x ∈ I for all x ∈ S [ t ] , λ ∈ k .Fix λ ∈ k and suppose that x ∈ S [ t ] is such that ( t − λ ) x ∈ I . Then by assumption ( t − λ ) x ∈ ( t − λ ) I + I [ t ] , so that ( t − λ )( x − i ) ∈ I [ t ] for some i ∈ I . Since S [ t ] /I [ t ] ≃ S/I [ t ] is a free k [ t ] -module, we have x − i ∈ I [ t ] ⊆ I and so x ∈ I . Remark 2.13.
Let i , . . . , i r be the generators of I . To check the inclusion which is the assump-tion of Proposition 2.12, it is enough to check that s ∈ ( t − λ ) ∩ I implies s ∈ ( t − λ ) I + I [ t ] forall s = s i + . . . + s r i r , where s i ∈ S .Indeed, take an arbitrary element s ∈ I and write s = t i + . . . + t r i r , where t , . . . , t r ∈ S [ t ] .Dividing t i by t − λ we obtain s = s i + . . . + s r i r + ( t − λ ) i , where i ∈ I and s i ∈ S . Denote s ′ = s i + . . . + s r i r , then s ∈ ( t − λ ) ∩ I if and only if s ′ ∈ ( t − λ ) ∩ I and s ∈ ( t − λ ) I + I [ t ] if and only if s ′ ∈ ( t − λ ) I + I [ t ] . Example 2.14.
Consider S = k [ x, y ] and I = xyS [ t ] + ( x − tx ) S [ t ] ⊆ S [ t ] . Take an element s xy + s ( x − tx ) ∈ I and suppose s xy + s ( x − tx ) ∈ ( t − λ ) S [ t ] . We want to prove thatthis element lies in I [ t ] + ( t − λ ) I . As in Remark 2.13, by subtracting an element of I ( t − λ ) we may assume that s , s lie in S . Then s xy + s ( x − tx ) ∈ ( t − λ ) S [ t ] if and only if s xy + s ( x − λx ) = 0 . In particular we have s ∈ yS so that s ( x − tx ) ∈ xyS [ t ] , then s xy + s ( x − tx ) ∈ xyS [ t ] ⊆ I [ t ] . Lemma 2.15.
Consider a ring R = B [ α ] graded by the degree of α . Let d be a natural numberand I ⊆ R be a homogeneous ideal generated in degrees less or equal to d .Let q ∈ B [ α ] be an element of α -degree strictly less than d and such that for every b ∈ B satisfying bα d ∈ I , we have bq ∈ I . Then for every r ∈ R the condition r ( α d − q ) ∈ I implies rα d ∈ I and rq ∈ I. Proof.
We apply induction with respect to α -degree of r , the base case being r = 0 . Write r = m X i =0 r i α i , where r i ∈ B. The leading form of r ( α d − q ) is r m α m + d and it lies in I . Since I is homogeneous and generatedin degree at most d , we have r m α d ∈ I . Then r m q ∈ I by assumption, so that ˆ r := r − r m α m satisfies ˆ r ( α d − q ) ∈ I . By induction we have ˆ rα d , ˆ rq ∈ I , then also rα d , rq ∈ I . Definition 3.1.
Let f ∈ P = k [ x , . . . , x n ] be a polynomial of degree s . Let I = ann S ( f ) and A = S/I = Apolar ( f ) . By ∆ • we denote the decomposition of the Hilbert function of A and weset e ( a ) := P at =0 ∆ t (1) .We say that f is in the standard form if f = f + f + f + f + · · · + f s , where f i ∈ P i ∩ k [ x , . . . , x e ( s − i ) ] for all i. Note that if f is in the standard form and ∂ ∈ m S then f + ∂ y f is also in the standard form.We say that an Artin Gorenstein algebra S/I is in the standard form if any (or every) dual soclegenerator of
S/I is in the standard form, see Proposition 3.5 below.
Example 3.2. If f = x + x + x , then f is in the standard form. Indeed, e (0) = 1 , e (1) = 2 , e (2) = 2 , e (3) = 3 so that we should check that x ∈ k [ x ] , x ∈ k [ x , x ] , x ∈ k [ x , x , x ] ,which is true. On the contrary, g = x + x + x is not in the standard form, but may be put inthe standard form via a change of variables. The change of variables procedure of Example 3.2 may be generalised to prove that everylocal Artin Gorenstein algebra can be put in a standard form, as the following Proposition 3.3explains.
Proposition 3.3.
For every Artin Gorenstein algebra
S/I there is an automorphism ϕ : S → S such that S/ϕ ( I ) is in the standard form.Proof. See [Iar94, Thm 5.3AB], the proof is rewritten in [Jel13, Thm 4.38].The idea of the proof of Proposition 3.3 is to “linearise” some elements of S . This is quitetechnical and perhaps it can be best seen on the following example. Example 3.4.
On this example we exhibit the proof of Proposition 3.3. Let f = x + x x . Theannihilator of f in S is ( α , α − α α ) , the Hilbert function of Apolar ( f ) is (1 , , , , , , and the symmetric decomposition is ∆ = (1 , , , , , , , ∆ = (0 , , , , , , ∆ = (0 , , , , . his shows that e (0) = 1 , e (1) = 1 , e (2) = 2 . If f is in the standard form we should have f = x x ∈ k [ x , . . . , x e (1) ] = k [ x ] . This means that f is not in the standard form. The“reason” for e (1) = 1 is the fact that α ( α − α ) annihilates f , and the “reason” for f k [ x ] isthat α − α is not a linear form. Thus we make α − α a linear form by twisting by a suitableautomorphism of S .We define an automorphism ψ : S → S by ψ ( α ) = α and ψ ( α ) = α + α , so that we have ψ ( α − α ) = α . The automorphism maps the annihilator of f to the ideal I := (( α + α ) , α α ) .We will see that the algebra S/I is in the standard form and also find a particular dual generatorobtained from f .As mentioned in Remark 2.2, the automorphism ψ induces an automorphism ψ ∗ of the k -linear space P ≤ . This automorphism maps f to a dual socle generator ψ ∗ f of S/I .The element F := ψ ∗ x is the only element of P such that ψ ( α ) y F = ψ ( α ) y F = 0 , ψ ( α )( F ) = 1 and ψ ( α l )( F ) = 0 for l ≤ . Caution: in the last line we use evaluation onthe functional and not the induced action (see Remark 2.2). One can compute that ψ ∗ x = x − x x + x x − x and similarly ψ ∗ x x = x x − x x + 3 x so that ψ ∗ f = x − x x + 2 x .Now indeed x ∈ k [ x ] , x x ∈ k [ x , x ] and x ∈ k [ x , x ] so the dual socle generator is in thestandard form. We note the following equivalent conditions for a dual socle generator to be in the standardform.
Proposition 3.5.
In the notation of Definition 3.1, the following conditions are equivalent fora polynomial f ∈ P :1. the polynomial f is in the standard form,2. for all r and i such that r > e ( s − i ) we have m i − S α r ⊆ I = ( f ) ⊥ . Equivalently, for all r and i such that r > e ( i ) we have m s − i − S α r ⊆ I = ( f ) ⊥ .Proof. Straightforward.
Corollary 3.6.
Let f ∈ P be such that the algebra S/I is in the standard form, where I =ann S ( f ) . Let ϕ be an automorphism of S given by ϕ ( α i ) = κ i α i + q i where q i is such that deg( q i y f ) ≤ deg( α i y f ) and κ i ∈ k \ { } . Then the algebra
S/ϕ − ( I ) is also in the standard form.Proof. The algebras
S/I and
S/ϕ − ( I ) are isomorphic, in particular they have equal functions e ( · ) . By Proposition 3.5 it suffices to prove that if for some r, i we have m rS α i ⊆ I , then m rS α i ⊆ ϕ − ( I ) . The latter condition is equivalent to m rS ϕ ( α i ) ⊆ I . If m rS α i y f = 0 then deg( α i y f ) < r so, by assumption, deg( q i y f ) < r thus m rS q i y f = 0 and m rS ϕ ( α i ) = m rS ( κ i α i + q i ) y f = 0 . Corollary 3.7.
Suppose that q ∈ m S does not contain α i and let ϕ : S → S be an automorphismgiven by ϕ ( α j ) = α j for all j = i and ϕ ( α i ) = κ i α i + q, where κ i ∈ k \ { } . Suppose that
S/I is in the standard form, where I = ann S ( f ) and that deg( q y f ) ≤ deg( α i y f ) .Then the algebras S/ϕ ( I ) and S/ϕ − ( I ) are also in the standard form.Proof. Note that ψ : S → S given by ψ ( α j ) = α j for j = i and ψ ( α i ) = κ − i ( α i − q ) is anautomorphism of S and furthermore ψ ( κ i α i + q ) = α i − q + q = α i so that ψ = ϕ − . Both ϕ and ψ satisfy assumptions of Corollary 3.6 so both S/ϕ − ( I ) and S/ψ − ( I ) = S/ϕ ( I ) are in thestandard form. 11 emark 3.8. The assumption q ∈ m S of Corollary 3.7 is needed only to ensure that ϕ is anautomorphism of S . On the other hand the fact that q does not contain α i is important, becauseit allows us to control ϕ − and in particular prove that S/ϕ ( I ) is in the standard form. The following Corollary 3.9 is a straightforward generalisation of Corollary 3.7, but the no-tation is difficult. We first choose a set K of variables. The automorphism sends each variablefrom K to (a multiple of) itself plus a suitable polynomial in variables not appearing in K . Corollary 3.9.
Take
K ⊆ { , , . . . , n } and q i ∈ m S for i ∈ K which do not contain any variablesfrom the set { α i } i ∈K . Define ϕ : S → S by ϕ ( α i ) = ( α i if i / ∈ K κ i α i + q i , where κ i ∈ k \ { } if i ∈ K . Suppose that
S/I is in the standard form, where I = ann S ( f ) and that deg( q i y f ) ≤ deg( α i y f ) for all i ∈ K . Then the algebras S/ϕ ( I ) and S/ϕ − ( I ) are also in the standard form. Recall that k is an algebraically closed field of characteristic neither nor .In the previous section we mentioned that for every local Artin Gorenstein algebra thereexists a dual socle generator in the standard form, see Definition 3.1. In this section we will seethat in most cases we can say more about this generator. Our main aim is to put the generatorin the form x s + f , where f contain no monomial divisible by a “high” power of x . We will useit to prove that families arising from certain ray decompositions (see Definition 5.2) are flat.We begin with an easy observation. Remark 4.1.
Suppose that a polynomial f ∈ P is such that H Apolar( f ) (1) equals the number ofvariables in P . Then any linear form in P is a derivative of f . If deg f > then the S -submodules Sf and S ( f − f − f ) are equal, so analysing this modules we may assume f = f = 0 , i.e. thelinear part of f is zero.Later we use this remark implicitly. The following Lemma 4.2 provides a method to slightly improve the given dual socle generator.This improvement is the building block of all other results in this section.
Lemma 4.2.
Let f ∈ P be a polynomial of degree s and A be the apolar algebra of f . Supposethat α s y f = 0 . For every i let d i := deg( α α i y f ) + 2 .Then A is isomorphic to the apolar algebra of a polynomial ˆ f of degree s , such that α s y ˆ f = 1 and α d i − α i y ˆ f = 0 for all i = 1 . Moreover, the leading forms of f and ˆ f are equal up to anon-zero constant. If f is in the standard form, then ˆ f is also in the standard form.Proof. By multiplying f by a non-zero constant we may assume that α s y f = 1 . Denote I :=ann S ( f ) . Since deg( α α i y f ) = d i − , the polynomial α d i − α i y f = α d i − ( α α i y f ) is constant;we denote it by λ i . Then (cid:16) α d i − α i − λ i α s (cid:17) y f = 0 , so that α d i − (cid:16) α i − λ i α s − d i +11 (cid:17) ∈ I. Define an automorphism ϕ : S → S by ϕ ( α i ) = ( α if i = 1 α i − λ i α s − d i +11 if i = 1 , α d i − α i ∈ ϕ − ( I ) for all i > . The dual socle generator ˆ f of the algebra S/ϕ − ( I ) has therequired form. We can easily check that the graded algebras of S/ϕ − ( I ) and S/I are equal, inparticular ˆ f and f have the same leading form, up to a non-zero constant.Suppose now that f is in the standard form. Let i ∈ { , . . . , n } . Then d i = deg( α α i y f )+2 ≤ deg( α i y f ) + 1 , so that deg( α s − d i +11 y f ) ≤ d i − ≤ deg( α i y f ) . Since ϕ is an automorphism of S ,by Remark 3.8 we may apply Corollary 3.9 to ϕ . Then S/ϕ ( I ) is in the standard form, so ˆ f isin the standard form by definition. Example 4.3.
Let f ∈ k [ x , x , x , x ] be a polynomial of degree s . Suppose that the leadingform f s of f can be written as f s = x s + g s where g s ∈ k [ x , x , x ] . Then deg( α α i y f ) ≤ s − for all i > . Using Lemma 4.2 we produce ˆ f = x s + h such that the apolar algebras of f and ˆ f are isomorphic and α s − α i y h = 0 for all i = 1 . Then α s − y h = λ x + λ , where λ i ∈ k for i = 1 , . After adding a suitable derivative to ˆ f , we may assume λ = λ = 0 , i.e. α s − y h = 0 . Example 4.4.
Suppose that a local Artin Gorenstein algebra A of socle degree s has Hilbertfunction equal to (1 , H , H , . . . , H c , , . . . , . The standard form of the dual socle generator of A is f = x s + κ s − x s − + · · · + κ c +2 x c +21 + g, where deg g ≤ c + 1 and κ • ∈ k . By adding a suitable derivative we may furthermore make all κ i = 0 and assume that α c +11 y g = 0 . Using Lemma 4.2 we may also assume that α c α j y g = 0 for every j = 1 so we may assume α c y g = 0 , arguing as in Example 4.3. This gives a dual soclegenerator f = x s + g, where deg g ≤ c + 1 and g does not contain monomials divisible by x c . The following proposition was proved in [CN13] under the assumption that k is algebraicallyclosed of characteristic zero and in [Jel13, Thm 5.1] under the assumption that k = C . Forcompleteness we include the proof (with no further assumptions on k other than the ones listedat the beginning of this section). Proposition 4.5.
Let A be Artin local Gorenstein algebra of socle degree s ≥ such that theHilbert function decomposition from Theorem 2.3 has ∆ A,s − = (0 , q, . Then A is isomorphicto the apolar algebra of a polynomial f such that f is in the standard form and the quadric part f of f is a sum of q squares of variables not appearing in f ≥ and a quadric in variables appearingin f ≥ .Proof. Let us take a standard dual socle generator f ∈ P := k [ x , . . . , x n ] of the algebra A . Nowwe will twist f to obtain the required form of f . We may assume that H Apolar( f ) (1) = n .If s = 2 , then the theorem follows from the fact that the quadric f may be diagonalised.Assume s ≥ . Let e := e ( s −
3) = P s − t =0 ∆ A,t (1) . We have n = e ( s −
2) = f + q , so that f ≥ ∈ k [ x , . . . , x e ] and f ∈ k [ x , . . . , x n ] . Note that f ≥ is also in the standard form, so thatevery linear form in x , . . . , x e is a derivative of f ≥ , see Remark 4.1.First, we want to assure that α n y f = 0 . If α n y f ∈ k [ x , . . . , x e ] then there exists an operator ∂ ∈ m S such that ( α n − ∂ ) y f = 0 . This contradicts the fact that f was in the standard form (seethe discussion in Example 3.4). So we get that α n y f contains some x r for r > e , i.e. f containsa monomial x r x n . A change of variables involving only x r and x n preserves the standard formand gives α n y f = 0 .Applying Lemma 4.2 to x n we see that f may be taken to be in the form ˆ f + x n , where ˆ f does not contain x n , i.e. ˆ f ∈ k [ x , . . . , x n − ] . We repeat the argument for ˆ f .13 xample 4.6. If A is an algebra of socle degree , then H A = (1 , n, e, for some n , e . Moreover, n ≥ e and the symmetric decomposition of H A is (1 , e, e,
1) + (0 , n − e, . By Proposition 4.5 wesee that A is isomorphic to the apolar algebra of f + X e
Let
R ⊆ H ilb r Spec S be a constructible subset and V ⊆ P denote the set ofall possible dual socle generators of elements of R . If R is irreducible, then also V is irreducible.Proof. Below by k ∗ and S ∗ we denote the sets of invertible elements of k and S respectively.There is an induced surjective morphism ϕ from V to R as explained above. By constructionthe fiber over ϕ ( f ) is S ∗ y f . The image R of ϕ is irreducible, so it is enough to show the existenceof an open cover { H i } of R such that every ϕ − ( H i ) is irreducible.Choose an element f ∈ V and a section of m S / ann S ( f ) to m S , that is, a linear subspace m ( f ) ⊆ m S such that m ( f ) → m S / ann S ( f ) is bijective. Let O ( f ) := m ( f ) + k ⊆ S , then S y f = O ( f ) y f . Finally let O ( f ) ∗ := k ∗ + m ( f ) , so that ϕ − ( ϕ ( f )) = O ( f ) ∗ y f . Consider the set U f = { g ∈ V | O ( f ) ∩ ann S ( g ) = 0 } = { g ∈ V | O ( f ) y g = S y g } . It is an open set in V and its image H f = ϕ ( U f ) is open (hence irreducible) in the Hilbertscheme. Moreover U f = ϕ − ( H f ) . For every g ∈ U f the fiber ϕ − ( ϕ ( g )) is equal to O ( f ) ∗ y g .By [Ems78, Proposition 18 and its Corollary] there is an open neighborhood H ′ f ⊆ H f of ϕ ( f ) such that the morphism ϕ : ϕ − ( H ′ f ) → H ′ f has a section i . Denoting ϕ − ( H ′ f ) by U ′ f , wehave a surjective morphism O ( f ) ∗ × H ′ f → U ′ f mapping ( σ, h ) to σ y i ( h ) . Since O ( f ) ∗ and H ′ f are irreducible, also U ′ f is irreducible. Therefore { H ′ f } form a desired cover of R and so V isirreducible. Proposition 4.8.
Let H = (1 , , , ∗ , . . . , ∗ , be a vector of length s + 1 . The set of polynomials f ∈ k [ x , x ] such that H Apolar( f ) = H constitutes an irreducible subscheme of the affine space k [ x , x ] ≤ s . A general member of this set has, up to an automorphism of P induced by anautomorphism of S , the form f + ∂ y f , where f = x s + x s for some s ≤ s .Proof. Let V ⊆ k [ x , x ] denote the set of f such that H Apolar( f ) = H . Then the image of V underthe mapping sending f to Apolar ( f ) is irreducible by [Iar77, Thm 3.13]. By Proposition 4.7 theset V is irreducible. 14n the case H = (1 , , , . . . , the claim (with s = 0 ) follows directly from the existence ofthe standard form of a polynomial. Further in the proof we assume H (1) = 2 .Let us take a general polynomial f such that H Apolar( f ) = H . Then ann S ( f ) = ( q , q ) is acomplete intersection, where q ∈ S has order , i.e. q ∈ m S \ m S . Since f is general, we mayassume that the quadric part of q has maximal rank, i.e. rank two, see also [Iar77, Thm 3.14].Then after a change of variables q ≡ α α mod m S . Since the leading form α α of q isreducible, q = δ δ for some δ , δ ∈ S such that δ i ≡ α i mod m S for i = 1 , , see e.g. [Kun05,Thm 16.6]. After an automorphism of S we may assume δ i = α i , then α α = q annihilates f ,so that it has the required form. It is well-known that if F ∈ P s is a form such that H Apolar( F ) = (1 , , . . . , , then the standardform of F is either x s + x s or x s − x . In particular the set of such forms in P is irreducible andin fact it is open in the so-called secant variety. This section is devoted to some generalisationsof this result for the purposes of classification of leading forms of polynomials in P .The following proposition is well-known if the base field is of characteristic zero (see [BGI11,Thm 4] or [LO13]), but we could not find a reference for the positive characteristic case, so forcompleteness we include the proof. Proposition 4.9.
Suppose that F ∈ k [ x , x , x ] is a homogeneous polynomial of degree s ≥ .The following conditions are equivalent1. the algebra Apolar ( F ) has Hilbert function H beginning with H (1) = H (2) = H (3) = 3 ,i.e. H = (1 , , , , . . . ) ,2. after a linear change of variables F is in one of the forms x s + x s + x s , x s − x + x s , x s − ( x x + x ) . Furthermore, the set of forms in k [ x , x , x ] s satisfying the above conditions is irreducible.Proof. For the characteristic zero case see [LO13] and references therein.Let S = k [ α , α , α ] be a polynomial ring dual to P . This notation is incoherent with theglobal notation, but it is more readable than S poly .Let I := ann S ( F ) and I := h θ , θ , θ i ⊆ S be the linear space of operators of degree annihilating F . Let A := S/I , J := ( I ) ⊆ S and B := S/J . Since A has length greater than · > , the ideal J is not a complete intersection. Let us analyse the Hilbert function of A .By symmetry of H A , we have H A ( s −
1) = H A (1) = 3 . By Remark 2.7 we have H A (3) ≥ H A (4) ≥ . . . ≥ H A ( s −
1) = 3 , thus H A ( m ) = 3 for all m = 1 , , . . . , s − . We will prove that the graded ideal J is saturated and defines a zero-dimensional scheme ofdegree in P = Proj S . First, H A (3) ≤ H B (3) ≤ by Macaulay’s Growth Theorem. If H B (3) = 4 then by Lemma 2.9 and Remark 2.10 we have H A (1) = 2 , a contradiction. We haveproved that H B (3) = 3 .Now we want to prove that H B (4) = 3 . By Macaulay’s Growth Theorem applied to H B (3) =3 we have H B (4) ≤ . If s > then H A (4) = 3 , so H B (4) ≥ . Suppose s = 4 . By Buchsbaum-Eisenbud result [BE77] we know that the minimal number of generators of I is odd. Moreover,we know that A n = B n for n < , thus the generators of I have degree two or four. Since I is nota complete intersection, there are at least two generators of degree , so H B (4) ≥ H A (4) + 2 = 3 .15rom H B (3) = H B (4) = 3 by Gotzmann’s Persistence Theorem we see that H B ( m ) = 3 for all m ≥ . Thus the scheme Γ := V ( J ) ⊆ Proj k [ α , α , α ] is finite of degree and J issaturated. In particular, the ideal J = I (Γ) is contained in I .We will use Γ to compute the possible forms of F , in the spirit of Apolarity Lemma, see[IK99, Lem 1.15]. There are four possibilities for Γ :1. Γ is a union of three distinct, non-collinear points. After a change of basis Γ = { [1 : 0 : 0] }∪{ [0 : 1 : 0] } ∪ { [0 : 0 : 1] } , then I = ( α α , α α , α α ) and F = x s + x s + x s .2. Γ is a union of a point and scheme of length two, such that h Γ i = P . After a change ofbasis I Γ = ( α , α α , α α ) , so that F = x s − x + x s .3. Γ is irreducible with support [1 : 0 : 0] and it is not a -fat point. Then Γ is Gorensteinand so Γ may be taken as the curvilinear scheme defined by ( α , α α , α α − α ) . Then,after a linear change of variables, F = x s − x + x x s − .4. Γ is a -fat point supported at [1 : 0 : 0] . Then I Γ = ( α , α α , α ) , so F = x s − ( λ x + λ x ) for some λ , λ ∈ k . But then there is a degree one operator in S annihilating F , acontradiction.The set of forms F which are sums of three powers of linear forms is irreducible. To see thatthe forms satisfying the assumptions of the Proposition constitute an irreducible subset of P s we observe that every Γ as above is smoothable by [CEVV09]. The flat family proving thesmoothability of Γ induces a family F t → F , such that F λ is a sum of three powers of linearforms for λ = 0 , see [Ems78, Corollaire in Section 2]. See also [BB14] for a generalisation of thismethod. Proposition 4.10.
Let s ≥ . Consider the set S of all forms F ∈ k [ x , x , x , x ] of degree s such that the apolar algebra of F has Hilbert function (1 , , , , . . . , , . This set is irreducibleand its general member has the form ℓ s + ℓ s + ℓ s + ℓ s , where ℓ , ℓ , ℓ , ℓ are linearly independentlinear forms.Proof. First, the set S of forms equal to ℓ + ℓ + ℓ + ℓ , where ℓ , ℓ , ℓ , ℓ are linearlyindependent linear forms, is irreducible and contained in S . Then, it is enough to prove that S lies in the closure of S .We follow the proof of Proposition 4.9, omitting some details which can be found there. Let S = k [ α , α , α , α ] , I := ann S ( F ) and J := ( I ) . Set A = S/I and B = S/J . Then H B (2) = 4 and H B (3) is either or . If H B (3) = 5 , then by Lemma 2.9 we have H B (1) = 3 , a contradiction.Thus H B (3) = 4 .Now we would like to prove H B (4) = 4 . By Macaulay’s Growth Theorem H B (4) ≤ . ByLemma 2.9 H B (4) = 5 , thus H B (4) ≤ . If s > then H B (4) ≥ H A (4) ≥ , so we concentrateon the case s = 4 . Let us write the minimal free resolution of A , which is symmetric by [Eis95,Cor 21.16]: → S ( − → S ( − ⊕ a ⊕ S ( − ⊕ → S ( − ⊕ b ⊕ S ( − ⊕ c ⊕ S ( − ⊕ b → S ( − ⊕ ⊕ S ( − ⊕ a → S. Calculating H A (3) = 4 from the resolution, we get b = 8 . Calculating H A (4) = 1 we obtain − a + c = 0 . Since a = H B (4) ≤ we have a ≤ , so a = 3 , c = 0 and H B (4) = 4 .Now we calculate H B (5) . If s > then H B (5) = 4 as before. If s = 4 then extracting syzygiesof I from the above resolution we see that H B (5) = 4 + γ , where ≤ γ ≤ , thus H B (5) = 4 and γ = 0 . If s = 5 , then the resolution of A is → S ( − → S ( − ⊕ ⊕ S ( − ⊕ → S ( − ⊕ ⊕ S ( − ⊕ → S ( − ⊕ ⊕ S ( − ⊕ → S. H B (5) = 56 − · . Thus, as in the previous case we see that J is the saturatedideal of a scheme Γ of degree . Then Γ is smoothable by [CEVV09] and its smoothing inducesa family F t → F , where F λ ∈ S for λ = 0 .The following Corollary 4.11 is a consequence of Proposition 4.10. This corollary is not usedin the proofs of the main results, but it is of certain interest of its own and shows anotherconnection with secant varieties. For simplicity and to refer to some results from [LO13], weassume that k = C , but the claim holds for all fields of characteristic either zero or large enough.To formulate the claim we introduce catalecticant matrices. Let ϕ a,s − a : S a × P s → P s − a bethe contraction mapping applied to homogeneous polynomials of degree s . For F ∈ P s we obtain ϕ a,s − a ( F ) : S a → P s − a , whose matrix is called the a -catalecticant matrix . It is straightforwardto see that rk ϕ a,s − a ( F ) = H Apolar( F ) ( a ) . Corollary 4.11.
Let s ≥ and k = C . The fourth secant variety to s -th Veronese reembeddingof P n is a subset σ ( v s ( P n )) ⊆ P ( P s ) set-theoretically defined by the condition rk ϕ a,s − a ≤ ,where a = ⌊ s/ ⌋ .Proof. Since H Apolar( F ) ( a ) ≤ for F which is a sum of four powers of linear forms, by semicon-tinuity every F ∈ σ ( v s ( P n )) satisfies the above condition.Let F ∈ P s be a form satisfying rk ϕ a,s − a ( F ) ≤ . Let A = Apolar ( F ) and H = H A be theHilbert function of A . We want to reduce to the case where H ( n ) = 4 for all < n < s .First we show that H ( n ) ≥ for all < n < s . If H (1) ≤ , then the claim followsfrom [LO13, Thm 3.2.1 (2)], so we assume H (1) ≥ . Suppose that for some n satisfying ≤ n < s we have H ( n ) < . Then by Remark 2.7 we have H ( m ) ≤ H ( n ) for all m ≥ n , so that H (1) = H ( s − < , a contradiction. Thus H ( n ) ≥ for all n ≥ . Moreover, H (3) ≥ byMacaulay’s Growth Theorem. Suppose now that H (2) < . By Theorem 2.6 the only possiblecase is H (2) = 3 and H (3) = 4 . But then H (1) = 2 < by Lemma 2.9, a contradiction. Thuswe have proved that H ( n ) ≥ for all < n < s. (1)We have H ( a ) = 4 . If s ≥ , then a ≥ , so by Remark 2.7 we have H ( n ) ≤ for all n > a .Then by the symmetry H ( n ) = H ( s − n ) we have H ( n ) ≤ for all n . Together with H ( n ) ≥ for < n < s , we have H ( n ) = 4 for < n < s . Then F ∈ σ ( v s ( P n )) by Proposition 4.10. If a = 3 (i.e. s = 6 or s = 7 ), then H (4) ≤ by Lemma 2.9 and we finish the proof as in the case s ≥ . If s = 5 , then a = 2 and the Hilbert function of A is (1 , n, , , n, . Again by Lemma 2.9,we have n ≤ , thus n = 4 by (1) and Proposition 4.10 applies. If s = 4 , then H = (1 , n, , n, .Suppose n ≥ , then Lemma 2.9 gives n ≤ , a contradiction. Thus n = 4 and Proposition 4.10applies also to this case.Note that for s ≥ the Corollary 4.11 was also proved, in the case k = C , in [BB14, Thm 1.1]. Recall that k is an algebraically closed field of characteristic neither nor . Since k [[ α i ]] is adiscrete valuation ring, all its ideals have the form α νi k [[ α i ]] for some ν ≥ . We use this propertyto construct certain decompositions of the ideals in the power series ring S . Definition 5.1.
Let I be an ideal of finite colength in the power series ring k [[ α , . . . , α n ]] and π i : k [[ α , . . . , α n ]] ։ k [[ α i ]] be the projection defined by π i ( α j ) = 0 for j = i and π i ( α i ) = α i .The i -th ray order of I is a non-negative integer ν = rord i ( I ) such that π i ( I ) = ( α νi ) . By the discussion above, the ray order is well-defined. Below by p i we denote the kernel of π i ; this is the ideal generated by all variables except for α i .17 efinition 5.2. Let I be an ideal of finite colength in the power series ring S = k [[ α , . . . , α n ]] .A ray decomposition of I with respect to α i consists of an ideal J ⊆ S , such that J ⊆ I ∩ p i ,together with an element q ∈ p i and ν ∈ Z + such that I = J + ( α νi − q ) S. Note that from Definition 5.1 it follows that for every I and i a ray decomposition (with J = I ∩ p i ) exists and that ν = rord i ( I ) for every ray decomposition. Definition 5.3.
Let S = k [[ α , . . . , α n ]] and S poly = k [ α , . . . , α n ] ⊆ S . Let I = J + ( α νi − q ) S be a ray decomposition of a finite colength ideal I ⊆ S . Let J poly = J ∩ S poly . The associated lower ray family is k [ t ] → S poly [ t ] J poly [ t ] + ( α νi − t · α i − q ) S poly [ t ] , and the associated upper ray family is k [ t ] → S poly [ t ] J poly [ t ] + ( α νi − t · α ν − i − q ) S poly [ t ] . If the lower (upper) family is flat over k [ t ] we will call it a lower (upper) ray degeneration . Note that the lower and upper ray degenerations agree for ν = 2 . Remark 5.4.
In all considered cases the quotient S poly /J poly will be finite over k , so that everyray family will be finite over k [ t ] . Then every ray degeneration will give a morphism to the Hilbertscheme. We leave this check to the reader. Remark 5.5.
In this remark for simplicity we assume that i = 1 in Definition 5.3. Below wewrite α instead of α . Let us look at the fibers of the upper ray family from this definition in aspecial case, when α · q ∈ J . The fiber over t = 0 is isomorphic to S/I . Let us take λ = 0 andanalyse the fiber at t = λ . This fiber is supported at (0 , , . . . , and at (0 , . . . , , λ, , . . . , ,where λ appears on the i -th position. In particular, this shows that the existence of an upperray degeneration proves that the algebra S/I is limit-reducible; this is true also for the lower raydegeneration.Now α ν +1 − λα ν is in the ideal defining the fiber of the upper ray family over t = λ . Nowone may compute that near (0 , . . . , the ideal defining the fiber is ( λα ν − − q ) + J . Similarlynear (0 , . . . , , λ, , . . . , it is ( α − λ ) + ( q ) + J . The argument is similar (though easier) to theproof of Proposition 5.10. Most of the families constructed in [CEVV09] and [CN09] are ray degenerations.
Definition 5.6.
For a non-zero polynomial f ∈ P and d ≥ the d -th ray sum of f with respectto a derivation ∂ ∈ m S is a polynomial g ∈ P [ x ] given by g = f + x d · ∂ y f + x d · ∂ y f + x d · ∂ y f + . . . . The following proposition shows that a ray sum naturally induces a ray decomposition, whichcan be computed explicitly.
Proposition 5.7.
Let g be the d -th ray sum of f with respect to ∂ ∈ m S such that ∂ y f = 0 . Let α be an element dual to x , so that P [ x ] and T := S [[ α ]] are dual. The annihilator of g in T isgiven by the formula ann T ( g ) = ann S ( f ) + d − X i =1 kα i ! ann S ( ∂ y f ) + ( α d − ∂ ) T, (2) where the sum denotes the sum of k -vector spaces. In particular, the ideal ann T ( g ) ⊆ T isgenerated by ann S ( f ) , α ann S ( ∂ y f ) and α d − ∂ . The formula (2) is a ray decomposition of ann T ( g ) with respect to α and with J = ann S ( f ) T + α ann S ( ∂ y f ) T and q = ∂ . roof. It is straightforward to see that the right hand side of Equation (2) lies in ann T ( g ) . Letus take any ∂ ′ ∈ ann T ( g ) . Reducing the powers of α using α d − ∂ we can write ∂ ′ = σ + σ α + · · · + σ k − α d − , where σ • do not contain α . The action of this derivation on g gives σ y f + xσ d − ∂ y f + x σ d − ∂ y f + · · · + x d − σ ∂ y f + x d ( . . . ) . We see that σ ∈ ann S ( f ) and σ i ∈ ann S ( ∂ y f ) for i ≥ , so the equality is proved. It is also clearthat J ⊆ m S T and ann T ( g ) = J + ( α d − ∂ ) T , so that indeed we obtain a ray decomposition. Remark 5.8.
It is not hard to compute the Hilbert function of the apolar algebra of a ray sumin some special cases. We mention one such case below. Let f ∈ P be a polynomial satisfying f = f = f = 0 and ∂ ∈ m S be such that ∂ y f = ℓ is a linear form, so that ∂ y f = 0 .Let A = Apolar ( f ) and B = Apolar (cid:0) f + x ℓ (cid:1) . The only different values of H A and H B are H B ( m ) = H A ( m ) + 1 for m = 1 , . The f = f = f = 0 assumption is needed to ensure thatthe degrees of ∂ y f and ∂ y ( f + x ℓ ) are equal for all ∂ not annihilating f . Proposition 5.9.
Let g be the d -th ray sum with respect to f and ∂ . Then the correspondingupper and lower ray families are flat. Recall, that these families are explicitly given as k [ t ] → T poly [ t ] J poly [ t ] + ( α d − tα d − − ∂ ) T poly [ t ] (upper ray family), (3) k [ t ] → T poly [ t ] J poly [ t ] + ( α d − tα − ∂ ) T poly [ t ] (lower ray family), (4) where T poly is the fixed polynomial subring of T .Proof. We start by proving the flatness of Family (4).We want to use Proposition 2.12. To simplify notation let J := J poly . Denote by I the idealdefining the family and suppose that some z ∈ I lies in ( t − λ ) for some λ ∈ k . Write z as i + i (cid:0) α d − tα − ∂ (cid:1) , where i ∈ J [ t ] , i ∈ T poly [ t ] , and note that by Remark 2.13 we may assume i ∈ J , i ∈ T poly . Since z ∈ ( t − λ ) , we have that i + i ( α d − λα − ∂ ) = 0 , so i ( α d − λα − ∂ ) = − i ∈ J. By Proposition 5.7 the ideal J is homogeneous with respect to grading by α . More precisely itis equal to J + J α , where J = ann S ( f ) T, J = ann S ( ∂ y f ) T are generated by elements notcontaining α , so that J is generated by elements of α -degree at most one. We now check theassumptions of Lemma 2.15. Note that ∂J ⊆ J by definition of J . If r ∈ T poly is such that rα d ∈ J , then r ∈ J , so that r ( λα + ∂ ) ∈ αJ + J ⊆ J . Therefore the assumptions are satisfiedand the Lemma shows that i α d ∈ J . Then i α ∈ J , thus i ( α d − tα ) ∈ J [ t ] ⊆ ( I ∩ T poly )[ t ] .Since i ∂ ∈ I ∩ T poly by definition, this implies that i + i ( α d − tα − ∂ ) ∈ J [ t ] ⊆ ( I ∩ T poly )[ t ] .Now the flatness follows from Proposition 2.12.The same proof works equally well for upper ray family: one should just replace α by α d − inappropriate places of the proof. For this reason we leave the case of Family (3) to the reader. Proposition 5.10.
Let us keep the notation of Proposition 5.9. Let λ ∈ k \ { } . The fibers ofthe Family (3) and Family (4) over t − λ are reducible.Suppose that ∂ y f = 0 and the characteristic of k does not divide d − . The fiber of theFamily (4) over t − λ is isomorphic to Spec Apolar ( f ) ⊔ (Spec Apolar ( ∂f )) ⊔ d − . roof. For both families the support of the fiber over t − λ contains the origin. The support ofthe fiber of Family (3) contains furthermore a point with α = λ and other coordinates equal tozero. The support of the fiber of Family (4) contains a point with α = ω , where ω d − = λ .Now let us concentrate on Family (4) and on the case ∂ y f = 0 . The support of the fiberover t − λ is (0 , . . . , , and (0 , . . . , , ω ) , where ω d − = λ are ( d − -th roots of λ , which arepairwise different because of the characteristic assumption. We will analyse the support pointby point. By hypothesis ∂ ∈ ann S ( ∂ y f ) , so that α · ∂ ∈ J , thus α d +1 − λ · α is in the ideal I ofthe fiber over t = λ .Near (0 , , . . . , the element α d − − λ is invertible, so α is in the localisation of the ideal I , thus α + λ − ∂ is in the ideal. Now we check that the localisation of I is equal to ann S ( f ) +( α + λ − ∂ ) T poly . Explicitly, one should check that (cid:0) ann S ( f ) + ( α + λ − ∂ ) T poly (cid:1) (0 ,..., = (cid:16) ann S ( f ) + ( α d − λα − ∂ ) T poly (cid:17) (0 ,..., . Then the stalk of the fiber at (0 , . . . , is isomorphic to Spec Apolar ( f ) .Near (0 , , . . . , , ω ) the elements α and α k +1 − λ · α α − ω are invertible, so ann S ( ∂ y f ) and α − ω are inthe localisation of I . This, along with the other inclusion, proves that this localisation is generatedby ann S ( ∂ y f ) and α − ω and thus the stalk of the fiber is isomorphic to Spec Apolar ( ∂f ) .We make the most important corollary explicit: Corollary 5.11.
We keep the notation of Proposition 5.9. Suppose that char k does not divide d − and ∂ y f = 0 . If both apolar algebras of f and ∂ y f are smoothable then also the apolaralgebra of every ray sum of f with respect to ∂ is smoothable. Example 5.12.
Let f ∈ k [ x , . . . , x n ] be a dual socle generator of an algebra A . Then thealgebra B = Apolar (cid:0) f + x n +1 (cid:1) is limit-reducible: it is a limit of algebras of the form A × k . Inparticular, if A is smoothable, then B is also smoothable.Combining this with Proposition 4.5, we see that every local Gorenstein algebra A of socledegree s with ∆ A,s − = (0 , q, , where q = 0 , is limit-reducible.If deg f ≥ , then the Hilbert functions of A = Apolar ( f ) and B = Apolar (cid:0) f + x n +1 (cid:1) arerelated by H B ( m ) = H A ( m ) for m = 1 and H B (1) = H A (1) + 1 . Above, we took advantage of the explicit form of ray decompositions coming from ray sumsto analyse the resulting ray families in depth. In Proposition 5.13 below we prove the flatness ofthe upper ray family without such knowledge. The price paid for this is the fact that we get noinformation about the fibers of this family.
Proposition 5.13.
Let f = x s + g ∈ P be a polynomial of degree s such that α c y g = 0 for some c satisfying c ≤ s . Then any ray decomposition ann S ( f ) = ( α ν − q ) + J , where J = ann S ( f ) ∩ ( α , . . . , α n ) , gives rise to an upper ray degeneration. In particular Apolar ( f ) islimit-reducible.Proof. Let I := ( α ν − tα ν − − q ) + J be the ideal defining the ray family and recall that q, J ⊆ p ,where p = ( α , . . . , α n ) .Since α ν − q ∈ ann S ( f ) , we have q y g = q y f = α ν y f = x s − ν + α ν y g . Then α s − ν ( q y g ) = α s − ν y x s − ν + α s y g = 1 , thus α s − ν y g = 0 . It follows that s − ν ≤ c − , so ν − ≥ s − c ≥ c , thus α ν − y g = 0 . For all γ ∈ p , we claim that γ · ( α ν − tα ν − − q ) ∈ J [ t ] . (5)Note that ( α ν − q ) y f = 0 and α ν − γ y f = α ν − γ y g = 0 . This means that α ν − γ ∈ J . Sincealways ( α ν − q ) γ ∈ J , we have proved (5). 20et I ⊆ S poly [ t ] be the ideal defining the upper ray family. Take any λ ∈ k and an element i ∈ I ∩ ( t − λ ) . We will prove that i ∈ I ( t − λ ) + I [ t ] , where I = I ∩ S , then Proposition 2.12asserts that S [ t ] / I is flat. Write i = i + i ( α ν − tα ν − − q ) . As before, we may assume i ∈ J , i ∈ S . Since i ∈ ( t − λ ) , we have i + i ( α ν − λα ν − − q ) = 0 . Since i ∈ p , we also have i ∈ p .But then by Inclusion (5) we have i ( α ν − tα ν − − q ) ⊆ I [ t ] . Since clearly i ∈ J ⊆ I [ t ] , theassumptions of Proposition 2.12 are satisfied, thus the upper ray family is flat.Now, Remark 5.5 shows that a general fiber of the upper ray degeneration is reducible, thus Apolar ( f ) is a flat limit of reducible algebras, i.e. limit-reducible. Example 5.14.
Let f ∈ k [ x , x , x , x ] be a polynomial of degree . Suppose that the leadingform f of f can be written as f = x + g where g ∈ k [ x , x , x ] . We will prove that Apolar ( f ) is limit-reducible. By Example 4.3 we may assume that f = x + g , where α y g = 0 .By Proposition 5.13 we see that Apolar ( f ) is limit-reducible. Example 5.15.
Suppose that an Artin local Gorenstein algebra A has Hilbert function H A =(1 , H , . . . , H c , , . . . , and socle degree s ≥ c . By Example 4.4 we may assume that A ≃ Apolar ( x s + g ) , where α c y g = 0 and deg g ≤ c + 1 . Then by Proposition 5.13 we obtain a flatdegeneration k [ t ] → S [ t ]( α ν − tα ν − − q ) + J . (6)
Thus A is limit-reducible in the sense of Definition 2.5. Let us take λ = 0 . By Remark5.5 the fiber over t = λ is supported at (0 , , . . . , and at ( λ, , . . . , and the ideal defin-ing this fiber near (0 , , . . . , is I = ( λα ν − − q ) + J . From the proof of 5.13 it followsthat α ν − y g = 0 . Then one can check that I lies in the annihilator of λ − x s − + g . Since σ y ( x s + g ) = σ y ( λ − x s − + g ) for every σ ∈ ( α , . . . , α n ) , one calculates that the apolar alge-bra of λ − x s − + g has Hilbert function (1 , H , . . . , H c , , . . . , and socle degree s − . Then dim k Apolar (cid:0) x s − + g (cid:1) = dim k Apolar (cid:0) λ − x s + g (cid:1) − . Thus the fiber is a union of a point and Spec Apolar (cid:0) λ − x s + g (cid:1) , i.e. degeneration (6) peels one point off A . A (finite) ray degeneration gives a morphism from
Spec k [ t ] to the Hilbert scheme, i.e. a curveon the Hilbert scheme H ilb ( P n ) . In this section we prove that in some cases the dimension ofthe tangent space to H ilb ( P n ) is constant along this curve. This enables us to prove that certainpoints of this scheme are smooth without the need for lengthy computations.This section seems to be the most technical part of the paper, so we include even moreexamples. The most important results here are Theorem 5.18 together with Corollary 5.20; seeexamples below Corollary 5.20 for applications.Recall (e.g. [Jel13, Prop 4.10] or [CN09]) that the dimension of the tangent space to H ilb ( P n ) at a k -point corresponding to a Gorenstein scheme Spec
S/I is dim k S/I − dim k S/I . Lemma 5.16.
Let d ≥ . Let g be the d -th ray sum of f ∈ P with respect to ∂ ∈ S such that ∂ y f = 0 . Denote I := ann S ( f ) and J := ann S ( ∂ y f ) . Take T = S [[ α ]] to be the ring dual to P [ x ] and let I := (cid:16) I + J α + ( α d − tα − ∂ ) (cid:17) · T [ t ] be the ideal in T [ t ] defining the associated lower ray degeneration, see Proposition 5.9. Then thefamily k [ t ] → T [ t ] / I is flat if and only if ( I : ∂ ) ∩ I ∩ J ⊆ I · J .Proof. To prove flatness we will use Proposition 2.12. Take an element i ∈ I ∩ ( t − λ ) . We wantto prove that i ∈ I ( t − λ ) + I [ t ] , where I [ t ] = I ∩ T . Let J := ( I + J α ) T . Subtracting asuitable element of I ( t − λ ) we may assume that i = i + i ( α d − tα − ∂ ) + i ( α d − tα − ∂ ) , i ∈ J , i ∈ J and i ∈ T . We will in fact show that i ∈ I ( t − λ ) + J [ t ] .To simplify notation denote σ = α d − λα − ∂ . Note that J σ ⊆ J . We have i + i σ + i σ = 0 .Let j := i σ . We want to apply Lemma 2.15, below we check its assumptions. The ideal J is homogeneous with respect to α , generated in degrees less than d . Let s ∈ T be an elementsatisfying sα d ∈ J . Then s ∈ J , which implies s ( λα + ∂ ) ∈ J . By Lemma 2.15 and i σ = j σ ∈ J we obtain j α d ∈ J , i.e. i σα d ∈ J . Applying the same argument to i α d we obtain i α d ∈ J , therefore i ∈ J T . Then i ( α d − tα − ∂ ) − i σ ( α d − tα − ∂ ) = i α ( t − λ )( α d − tα − ∂ ) ∈ J ( t − λ )( α d − tα − ∂ ) ⊆ I ( t − λ ) . Subtracting this element from i and substituting i := i + i σ we may assume i = 0 . We obtain i + i σ = i + i ( α d − λα − ∂ ) . (7)Let i = j + v α , where j ∈ S , i.e. it does not contain α . Since i ∈ J , we have j ∈ I . Asbefore, we have v α (( α d − tα − ∂ ) − σ ) = v α ( t − λ ) ∈ I ( t − λ ) , so that we may assume v = 0 .Comparing the top α -degree terms of (7) we see that j ∈ J . Comparing the terms of (7)not containing α , we deduce that j ∂ ∈ I , thus j ∈ ( I : ∂ ) . Jointly, j ∈ I ∩ J ∩ ( I : ∂ ) , thus j ∈ IJ by assumption. But then j α ∈ J , thus j ( α d − tα − ∂ ) ∈ J [ t ] and since i ∈ J , theelement i lies in J [ t ] ⊆ I [ t ] . Thus the assumptions of Proposition 2.12 are satisfied and thefamily T [ t ] / I is flat over k [ t ] .The converse is easier: one takes i ∈ I ∩ J ∩ ( I : ∂ ) such that i IJ . On one hand, theelement j := i ( α d − ∂ ) lies in J and we get that i ( α d − tα − ∂ ) − j = ti α ∈ I . On the otherhand if i α ∈ I , then i α ∈ ( I + ( t )) ∩ T = ( J + ( α d − ∂ )) , which is not the case. Remark 5.17.
Let us keep the notation of Lemma 5.16. Fix λ ∈ k \ { } and suppose thatthe characteristic of k does not divide d − . The supports of the fibers of S [ t ] / I , I / I and S [ t ] / I over t = λ are finite and equal. In particular from Proposition 5.10 it follows that thedimension of the fiber of I / I over t − λ is equal to tan( f ) + ( d −
1) tan( ∂ y f ) , where tan( h ) =dim k ann S ( h ) / ann S ( h ) is the dimension of the tangent space to the point of the Hilbert schemecorresponding to Spec S/ ann S ( h ) . Theorem 5.18.
Suppose that a polynomial f ∈ P corresponds to a smoothable, unobstructedalgebra Apolar ( f ) . Let ∂ ∈ S be such that ∂ y f = 0 and the algebra Apolar ( ∂ y f ) is smoothableand unobstructed. The following are equivalent:1. the d -th ray sum of f with respect to ∂ is unobstructed for some d such that ≤ d ≤ char k (or ≤ d if char k = 0 ).1a. the d -th ray sum of f with respect to ∂ is unobstructed for all d such that ≤ d ≤ char k (or ≤ d if char k = 0 ).2. The k [ t ] -module I / I is flat, where I is the ideal defining the lower ray family of the d -thray sum for some ≤ d ≤ char k (or ≤ d if char k = 0 ), see Definition 5.3.2a. The k [ t ] -module I / I is flat, where I is the ideal defining the lower ray family of the d -thray sum for every ≤ d ≤ char k (or ≤ d if char k = 0 ), see Definition 5.3.3. The family k [ t ] → S [ t ] / I is flat, where I is the ideal defining the lower ray family of the d -th ray sum for some ≤ d ≤ char k (or ≤ d if char k = 0 ).3a. The family k [ t ] → S [ t ] / I is flat, where I is the ideal defining the lower ray family of the d -th ray sum for every ≤ d ≤ char k (or ≤ d if char k = 0 ). . The following inclusion (equivalent to equality) of ideals in S holds: I ∩ J ∩ ( I : ∂ ) ⊆ I · J ,where I = ann S ( f ) and J = ann S ( ∂ y f ) .Proof. It is straightforward to check that the inclusion I · J ⊆ I ∩ J ∩ ( I : ∂ ) ⊆ I · J in Point4 always holds, thus the other inclusion is equivalent to equality.3. ⇐⇒ ⇐⇒ d , the equivalence of Point 4 and Point 3a also follows.2. ⇐⇒
3. and 2a. ⇐⇒ k [ t ] -modules → I / I → S [ t ] / I → S [ t ] / I → . Since S [ t ] / I is a flat k [ t ] -module by Proposition 5.9, we see from the long exact sequence of Tor that I / I is flat if and only if S [ t ] / I is flat.1. ⇐⇒
2. and 1a. ⇐⇒ g ∈ P [ x ] be the d -th ray sum of f with respect to ∂ .We may consider Apolar ( g ) , Apolar ( f ) , Apolar ( ∂ y f ) as quotients of a polynomial ring T poly ,corresponding to points of the Hilbert scheme. The dimension of the tangent space at Apolar ( g ) is given by dim k I / I ⊗ k [ t ] /t = dim k I / ( I + ( t )) . By Remark 5.17 it is equal to the sum ofthe dimension of the tangent space at Apolar ( f ) and ( d − times the dimension of the tangentspace to Apolar ( ∂ y f ) . Since both algebras are smoothable and unobstructed we conclude that Apolar ( g ) is also unobstructed. On the other hand, if Apolar ( g ) is unobstructed, then I / I is afinite k [ t ] -module such that the length of the fiber I / I ⊗ k [ t ] / m does not depend on the choiceof the maximal ideal m ⊆ k [ t ] . Then I / I is flat by [Har77, Ex II.5.8] or [Har77, Thm III.9.9]applied to the associated sheaf. Remark 5.19.
The condition from Point 4 of Theorem 5.18 seems very technical. It is enlight-ening to look at the images of ( I : ∂ ) ∩ I and I · J in I/I . The image of ( I : ∂ ) ∩ I is theannihilator of ∂ in I/I . This annihilator clearly contains ( I : ∂ ) · I/I = J · I/I . This shows thatif the S/I -module
I/I is “nice”, for example free, we should have an equality ( I : ∂ ) ∩ I = I · J .More generally this equality is connected to the syzygies of I/I . In the remainder of this subsection we will prove that in several situations the conditions ofTheorem 5.18 are satisfied.
Corollary 5.20.
We keep the notation and assumptions of Theorem 5.18. Suppose furtherthat the algebra
S/I = Apolar ( f ) is a complete intersection. Then the equivalent conditions ofTheorem 5.18 are satisfied.Proof. Since
S/I is a complete intersection, the
S/I -module
I/I is free, see e.g. [Mat86,Thm 16.2] and discussion above it or [Eis95, Ex 17.12a]. It implies that ( I : ∂ ) ∩ I = ( I : ∂ ) I = J I , because J = ann S ( ∂ y f ) = { s ∈ S | s∂ y f = 0 } = (ann S ( f ) : ∂ ) = ( I : ∂ ) . Thus thecondition from Point 4 of Theorem 5.18 is satisfied. Example 5.21. If A = S/I is a complete intersection, then it is smoothable and unobstructed(see Subsection 2.4). The apolar algebras of monomials are complete intersections, therefore theassumptions of Theorem 5.18 are satisfied e.g. for f = x x x and ∂ = α . Now Corollary 5.20implies that the equivalent conditions of the Theorem are also satisfied, thus x x x + x d x x =( x x )( x + x d ) is unobstructed for every d ≥ (provided char k = 0 or d ≤ char k ). Similarly, x x x + x x is unobstructed and has Hilbert function (1 , , , , . Example 5.22.
Let f = ( x + x ) x , then ann S ( f ) = ( α − α , α α , α ) is a complete inter-section. Take ∂ = α α , then ∂ y f = x and ∂ y f = 0 , thus f + x ∂ y f = x x + x x + x x isunobstructed. Note that by Remark 5.8 the apolar algebra of this polynomial has Hilbert function (1 , , , . roposition 5.23. Let f ∈ P be such that Apolar ( f ) is a complete intersection.Let d be a natural number. Suppose that char k = 0 or d ≤ char k . Take ∂ ∈ S such that ∂ y f = 0 and Apolar ( ∂ y f ) is also a complete intersection. Let g ∈ P [ y ] be the d -th ray sum f with respect to ∂ , i.e. g = f + y d ∂ y f .Suppose that deg ∂ y f > . Let β be the variable dual to y and σ ∈ S be such that σ y ( ∂ y f ) = 1 .Take ϕ := σβ ∈ T = S [[ β ]] . Let h be any ray sum of g with respect to ϕ , explicitly h = f + y d ∂ y f + z m y d − for some m ≥ .Then the algebra Apolar ( h ) is unobstructed.Proof. First note that ϕ y g = y d − and so ϕ y g = σ y y d − = 0 , since σ ∈ m S . Therefore indeed h has the presented form.From Corollary 5.20 it follows that Apolar ( g ) is unobstructed. Since ϕ y g = y d − , the algebra Apolar ( ϕ y g ) is unobstructed as well. Now by Theorem 5.18 it remains to prove that ( I g : ϕ ) ∩ I g ∩ J g ⊆ I g J g , (8)where I g = ann T ( g ) , J g = ann T ( ϕ y g ) . The rest of the proof is a technical verification of thisclaim. Denote I f := ann S ( f ) and J f := ann S ( ∂ y f ) ; note that we take annihilators in S . ByProposition 5.7 we have I g = I f T + βJ f T + ( β d − ∂ ) T . Consider γ ∈ T lying in ( I g : ϕ ) ∩ I g ∩ J g .Write γ = γ + γ β + γ β + . . . where γ i ∈ S , so they do not contain β . We will prove that γ ∈ I g J g .First, since ( β d − ∂ ) ∈ I g J g we may reduce powers of β in γ using this element and so weassume γ i = 0 for i ≥ d . Let us take i < d . Since γ ∈ J g = (cid:0) ann T (cid:0) y d − (cid:1)(cid:1) = (cid:0) m S , β d (cid:1) we seethat γ i ∈ m S ⊆ J g . For i > d we have β i ∈ I g , so that γ i β i ∈ J g I g and we may assume γ i = 0 .Moreover, β d γ d − ∂γ d ∈ I g J g so we may also assume γ d = 0 , obtaining γ = γ + · · · + γ d − β d − . From the explicit description of I g in Proposition 5.7 it follows that γ i ∈ J f for all i .Let M = I g ∩ ϕT = I g ∩ J f βT . Then for γ as above we have γϕ ∈ M , so we will analyse themodule M . Recall that I g = I f · T + βI f J f · T + β J f · T + ( β d − ∂ ) I f · T + ( β d − ∂ ) βJ f · T + ( β d − ∂ ) · T. (9)We claim that M ⊆ I f · T + βI f J f · T + β J f · T + ( β d − ∂ ) βJ f · T. (10)We have I g ⊆ J f · T + ( β d − ∂ ) · T , so if an element of I g lies in J f · T , then its coefficientstanding next to ( β d − ∂ ) in Presentation (9) is an element of J f by Lemma 2.15. Since J f · ( β d − ∂ ) ⊆ I f + βJ f , we may ignore the term ( β d − ∂ ) : M ⊆ I f · T + βI f J f · T + β J f · T + ( β d − ∂ ) I f · T + ( β d − ∂ ) βJ f · T. (11)Choose an element of M and let i ∈ I f · T be the coefficient of this element standing next to ( β d − ∂ ) . Since I f T ∩ βT ⊆ J f T we may assume that i does not contain β , i.e. i ∈ I f . Now, ifan element of the right hand side of (11) lies in β · T , then the coefficient i satisfies i · ∂ ∈ I f ,so that i ∈ ( I f : ∂ ) . Since I f is a complete intersection ideal the S/I f -module I f /I f is free, seeCorollary 5.20 for references. Then we have ( I f : ∂ ) = ( I f : ∂ ) I f and i ∈ ( I f : ∂ ) I f = I f J f .Then i · ( β d − ∂ ) ⊆ I f + β · I f · J f and so the Inclusion (10) is proved. We come back to the proofof proposition. 24rom Lemma 2.15 applied to the ideal J f T and the element β ( β d − ∂ ) and the fact that β∂J f ⊆ I g we compute that M ∩ { δ | deg β δ ≤ d } is a subset of I f · T + β · I f J f · T + β J f · T .Then γϕ = γβσ lies in this set, so that γ ∈ ( I f J f : σ ) and γ n ∈ ( J f : σ ) for n > . Since Apolar ( f ) and Apolar ( ∂ y f ) are complete intersections, we have γ ∈ I f m S and γ i ∈ J f m S for i ≥ . It follows that γ ∈ I g m S ⊆ I g J g . Example 5.24.
Let f ∈ P be a polynomial such that A = Apolar ( f ) is a complete intersection.Take ∂ such that ∂ y f = x and ∂ y f = 0 . Then the apolar algebra of f + y d x + y m y d − isunobstructed for any d, m ≥ (less or equal to char k if it is non-zero). In particular g = f + y x + y y is unobstructed.Continuing Example 5.22, if f = x x + x x , then x x + x x + x x + x x is unobstructed.The apolar algebra of this polynomial has Hilbert function (1 , , , .Let g = x x + x x + x x , then x x + x x + x x + x x is a ray sum of g with respect to ∂ = α α . Let I := ann S ( g ) and J := ( I : ∂ ) . In contrast with Corollary 5.20 and Example 5.22one may check that all three terms I , J and ( I : ∂ ) are necessary to obtain equality in theinclusion (8) for g and ∂ , i.e. no two ideals of I , J , ( I : ∂ ) have intersection equal to IJ . Example 5.25.
Let f = x + x . Then the annihilator of f in k [ α , α ] is a complete intersection,and this is true for every f ∈ k [ x , x ] . Let g = f + x x be the second ray sum of f with respectto α and h = g + x x be the second ray sum of g with respect to α α . Then the apolar algebraof h = x + x + x x + x x is smoothable and not obstructed. It has Hilbert function (1 , , , , , . Remark 5.26.
The assumption deg ∂ y f > in Proposition 5.23 is necessary: the polynomial h = x x x + x + x x is obstructed, with length and tangent space dimension > · over k = C . The polynomial g is the fourth ray sum of x x x with respect to α α α and h is thesecond ray sum of g = x x x + x with respect to α , thus this example satisfies the assumptionsof Proposition 5.23 except for deg ∂ y f > . Note that in this case α y g = 0 . Let r ≥ be a natural number and V be a constructible subset of P ≤ s . Assume that the apolaralgebra Apolar ( f ) has length r for every closed point f ∈ V . Then we may construct theincidence scheme { ( f, Apolar ( f )) } → V which is a finite flat family over V and thus we obtain amorphism from V to the (punctual) Hilbert scheme of r points on an appropriate P n . See [Jel13,Prop 4.39] for details.Consider f ∈ P ≤ s . The apolar algebra of f has length at most r if and only if the matrixof partials S ≤ s f has rank at most r . This is a closed condition, so we obtain the followingRemark 6.1. Remark 6.1.
Let s be a positive integer and V ⊆ P ≤ s be a constructible subset. Then the set U , consisting of f ∈ V such that the apolar algebra of f has the maximal length (among theelements of V ), is open in V . In particular, if V is irreducible then U is also irreducible. Example 6.2.
Let P ≥ = k [ x , . . . , x n ] ≥ be the space of polynomials that are sums of monomialsof degree at least . Suppose that the set V ⊆ P ≥ parameterising algebras with fixed Hilbertfunction H is irreducible. Then also the set W of polynomials f ∈ P such that f ≥ ∈ V isirreducible. Let e := H (1) and suppose that the symmetric decomposition of H has zero rows ∆ s − = (0 , , , and ∆ s − = (0 , , , where s = deg f . We claim that general element of corresponds to an algebra B with Hilbert function: H max = H + (0 , n − e, n − e, . Indeed,since we may only vary the degree three part of the polynomial, the function H B has the form H + (0 , a, a,
0) + (0 , b, for some a, b such that a + b ≤ n − e . Therefore algebras with Hilbertfunction H max are precisely the algebras of maximal possible length. Since H max is attained for f ≥ + x e +1 + . . . + x n , the claim follows from Remark 6.1. In the following H A denotes the Hilbert function of an algebra A . Lemma 6.3.
Suppose that A is a local Artin Gorenstein algebra of socle degree s ≥ such that ∆ A,s − = (0 , , . Then len A ≥ H A (1) + 1) . Furthermore, equality occurs if and only if s = 3 .Proof. Consider the symmetric decomposition ∆ • = ∆ A, • of H A . From symmetry we have P j ∆ ( j ) ≥ (1) with equality only if ∆ has no terms between and s − i.e. when s = 3 . Similarly P j ∆ i ( j ) ≥ i (1) for all ≤ i < s − . Summing these inequalities we obtain len A = X i
Let A be a local Artin Gorenstein algebra of length at most . Suppose that ≤ H A (1) ≤ . Then H A (2) ≤ .Proof. Let s be the socle degree of A . Suppose H A (2) ≥ . Then H A (3) + H A (4) + · · · ≤ , thus s ∈ { , , } . The cases s = 3 and s = 5 immediately lead to contradiction – it is impossibleto get the required symmetric decomposition. We will consider the case s = 4 . In this case H A = (1 , ∗ , ∗ , ∗ , and its symmetric decomposition is (1 , e, q, e,
1) + (0 , m, m,
0) + (0 , t, . Then e = H A (3) ≤ − − − . Since H A (1) < H A (2) we have e < q . This can only happen if e = 2 and q = 3 . But then ≥ len A = 9 + 2 m + t , thus m ≤ and H A (2) = m + q ≤ . Acontradiction. Lemma 6.5.
There does not exist a local Artin Gorenstein algebra with Hilbert function (1 , , , , , . . . , . Proof.
See [Iar94, pp. 99-100] for the proof or [CJN13, Lem 5.3] for a generalisation. We providea sketch for completeness. Suppose such an algebra A exists and fix its dual socle generator f ∈ k [ x , . . . , x ] s in the standard form. Let I = ann S ( f ) . The proof relies on two observations.First, the leading term of f is, up to a constant, equal to x s and in fact we may take f = x s + f ≤ . Moreover from the symmetric decomposition it follows that the Hilbert functions of Apolar ( x s + f ) and Apolar ( f ) are equal. Second, h (3) = 4 = 3 h i = h (2) h i is the maximalgrowth, so arguing similarly as in Lemma 2.9 we may assume that the degree two part I of theideal of gr A is equal to (( α , α ) S ) . Then any derivative of α y f is a derivative of x s , i.e. apower of x . It follows that α y f itself is a power of x ; similarly α y f is a power of x . Itfollows that f ∈ x · k [ x , x , x , x ] + k [ x , x ] , but then f is annihilated by a linear form, whichcontradicts the fact that f is in the standard form.The following lemmas essentially deal with the limit-reducibility in the case (1 , , , , , .Here the method is straightforward, but the cost is that the proof is broken into several casesand quite long. Lemma 6.6.
Let f = x + f be a polynomial such that H Apolar( f ) (2) < H Apolar( f ) (2) . Let Q = S ∩ ann S (cid:0) x (cid:1) ⊆ S . Then x ∈ Q f and ann S ( f ) ⊆ Q . roof. Note that dim Q f ≥ dim S f − H Apolar( f ) (2) − . If ann S ( f )
6⊆ Q , then thereis a q ∈ Q such that α − q ∈ ann S ( f ) . Then Q f = S f and we obtain a contradiction.Suppose that x
6∈ Q f . Then the degree two partials of f contain a direct sum of kx and Q f , thus they are at least H Apolar( f ) (2) -dimensional, so that H Apolar( f ) (2) ≥ H Apolar( f ) (2) , acontradiction. Lemma 6.7.
Let f = x + f ∈ P be a polynomial such that H Apolar( f ) = (1 , , , , , and H Apolar( f ) = (1 , , , , . Suppose that α y f = 0 and that (ann S ( f )) defines a completeintersection. Then Apolar ( f ) and Apolar ( f ) are complete intersections.Proof. Let I := ann S ( f ) . First we will prove that ann S ( f ) = ( q , q , c ) , where h q , q i = I and c ∈ I . Then of course Apolar ( f ) is a complete intersection. By assumption, q , q form aregular sequence. Thus there are no syzygies of degree at most three in the minimal resolutionof Apolar ( f ) . By the symmetry of the minimal resolution, see [Eis95, Cor 21.16], there are nogenerators of degree at least four in the minimal generating set of I . Thus I is generated indegree two and three. But H S/ ( q ,q ) (3) = 4 = H S/I (3) + 1 , thus there is a cubic c , such that I = kc ⊕ ( q , q ) , then ( q , q , c ) = I , thus Apolar ( f ) = S/I is a complete intersection.Let Q := ann S (cid:0) x (cid:1) ∩ S ⊆ S . By Lemma 6.6 we have q , q ∈ Q , so that α ∈ I \ ( q , q ) ,then I = ( q , q , α ) . Moreover, by the same Lemma, there exists σ ∈ Q such that σ y f = x .Now we prove that Apolar ( f ) is a complete intersection. Let J := ( q , q , α − σ ) ⊆ ann S ( f ) .We will prove that S/J is a complete intersection. Since q , q , α is a regular sequence, theset S/ ( q , q ) is a cone over a scheme of dimension zero and α does not vanish identically onany of its components. Since σ has degree two, α − σ also does not vanish identically on anyof the components of Spec S/ ( q , q ) , thus Spec
S/J has dimension zero, so it is a completeintersection (see also [VV78, Cor 2.4, Rmk 2.5]). Then the quotient by J has length at most deg( q ) deg( q ) deg( α − σ ) = 12 = dim k S/ ann S ( f ) . Since J ⊆ ann S ( f ) , we have ann S ( f ) = J and Apolar ( f ) is a complete intersection. Lemma 6.8.
Let f = x + f + g , where deg g ≤ , be a polynomial such that H Apolar ( f ≥ ) =(1 , , , , , and H Apolar( f ) = (1 , , , , . Suppose that α y f = 0 and that (ann S ( f )) doesnot define a complete intersection. Then Apolar ( f ) is limit-reducible.Proof. Let h q , q i = (ann S ( f )) . Since q , q do not form a regular sequence, we have, after alinear transformation ϕ , two possibilities: q = α α and q = α α or q = α and q = α α .Let β be the image of α under ϕ , so that β y f = 0 .Suppose first that q = α α and q = α α . If β is up to constant equal to α , then α α , α α , α ∈ ann S ( f ) , so that α is in the socle of Apolar ( f ) , a contradiction. Thus wemay assume, after another change of variables, that β = α , q = α α and q = α α . Then f = x + f + ˆ g = x + x + ˆ h + ˆ g , where ˆ h ∈ k [ x , x ] and deg(ˆ g ) ≤ . Then by Lemma 4.2we may assume that α y f = 0 , so Apolar ( f ) is limit-reducible by Proposition 5.13. See alsoExample 5.14 (the degree assumption in the Example can easily be modified).Suppose now that q = α and q = α α . If β is not a linear combination of α , α , thenwe may assume β = α . Let m in f be any monomial divisible by x . Since q , q ∈ ann S ( f ) ,we see that m = λx x for some λ ∈ k . But since β ∈ ann S ( f ) , we have m = 0 . Thus f doesnot contain x , so H Apolar( f ) (1) < , a contradiction. Thus β ∈ h α , α i . Suppose β = λα forsome λ ∈ k \ { } . Applying Lemma 6.6 to f ≥ we see that x is a derivative of f , so β y f = 0 ,but β y f = λ q y f = 0 , a contradiction. Thus β = λ α + λ α and changing α we mayassume that β = α . This substitution does not change h α , α α i . Now we directly check that f = x ( κ x x + κ x + κ x x + κ x ) , for some κ • ∈ k . Since x is a derivative of f , wehave κ = 0 . Then a non-zero element κ α α − κ α annihilates f . A contradiction with H Apolar( f ) (2) = 4 . 27 emma 6.9. Let a quartic f be such that H Apolar( f ) = (1 , , , , and α y f = 0 . Then H Apolar ( x + f )(2) ≥ .Proof. Let Q = ann S (cid:0) x (cid:1) ⊆ S . Let I denote the apolar ideal of f . By Proposition 4.9 wesee that I is minimally generated by three elements of degree two and two elements of degreefour. In particular, there are no cubics in the generating set. Since α ∈ I , there is an elementin σ ∈ I such that σ = α − q , where q ∈ Q . Therefore Q y f = S y f . Moreover, σ does notannihilate x , so that x is not a partial of f . We see that x and Q y f are leading forms ofpartials of x + f , thus H Apolar ( x + f )(2) ≥ Q y f ) = 1 + dim( S y f ) = 1 + H Apolar( f ) (2) = 4 . Remark 6.10.
In the setting of Lemma 6.9, it is not hard to deduce that H Apolar ( x + f ) =(1 , , , , , by analysing the possible symmetric decompositions. We do not need this strongerstatement, so we omit the proof. The following Proposition 6.11 generalises results about algebras with Hilbert function (1 , , , ,obtained in [Jel14] and [BCR12]. Proposition 6.11.
Let A be a local Artin Gorenstein algebra of socle degree three and H A (2) ≤ .Then A is smoothable.Proof. Suppose that the Hilbert function of A is (1 , n, e, . By Proposition 4.5 the dual soclegenerator of A may be put in the form f + x e +1 + · · · + x n , where f ∈ k [ x , . . . , x e ] . By repeateduse of Example 5.12 we see that A is a limit of algebras of the form Apolar ( f ) × k ⊕ n − e . Thusit is smoothable if and only if B = Apolar ( f ) is.Let e := H A (2) , then H B = (1 , e, e, . If H B (1) = e ≤ then B is smoothable. It remains toconsider ≤ e ≤ . The set of points corresponding to algebras with Hilbert function (1 , e, e, isirreducible in H ilb e +2 ( P e ) by Remark 6.1 for obvious parameterisation (as mentioned in [Iar84,Thm I, p. 350]), thus it will be enough to find a smooth point in this set which corresponds to asmoothable algebra. The cases e = 4 and e = 5 are considered in Example 5.22 and Example 5.24respectively. Remark 6.12.
The claim of Proposition 6.11 holds true if we replace the assumption H A (2) ≤ by H A (2) = 7 , thanks to the smoothability of local Artin Gorenstein algebras with Hilbert function (1 , , , , see [BCR12]. We will not use this result. Lemma 6.13.
Let A be a local Artin Gorenstein algebra with Hilbert function H A beginning with H A (0) = 1 , H A (1) = 4 , H A (2) = 5 , H A (3) ≤ . Then A is smoothable.Proof. Let f be a dual socle generator of A in the standard form. From Macaulay’s GrowthTheorem it follows that H A ( m ) ≤ for all m ≥ , so that H A = (1 , , , , , . . . , , , . . . , . Let s be the socle degree of A .Let ∆ A,s − = (0 , q, be the ( s − -nd row of the symmetric decomposition of H A . If q > ,then by Example 5.12 we know that A is limit-reducible; it is a limit of algebras of the form B × k ,such that H B (1) = H A (1) − . Then the algebra B is smoothable (see [CN09, Prop 2.5]), so A is also smoothable. In the following we assume that q = 0 .We claim that f ≥ ∈ k [ x , x ] . Indeed, the symmetric decomposition of the Hilbert functionis either (1 , , . . . ,
1) + (0 , , . . . , ,
0) + (0 , , , ,
0) + (0 , , , or (1 , , . . . , ,
1) + (0 , , , ,
0) +(0 , , , . In particular e ( s −
3) = P i ≥ ∆ i (1) = 2 , so that f ≥ ∈ k [ x , x ] and H Apolar ( f ≥ )(1) = , in particular x is a derivative of f ≥ , i.e. there exist a ∂ ∈ S such that ∂ y f ≥ = x . Then wemay assume ∂ ∈ m S , so ∂ y f = 0 .Let us fix f ≥ and consider the set of all polynomials of the form h = f ≥ + g , where g ∈ k [ x , x , x , x ] has degree at most three. By Example 6.2 the apolar algebra of a generalsuch polynomial will have Hilbert function H A . The set of polynomials h with fixed h ≥ = f ≥ ,such that H Apolar( h ) = H A , is irreducible. This set contains h := f ≥ + x x + x x . To finishthe proof is it enough to show that h is smoothable and unobstructed. Since Apolar ( f ≥ ) is acomplete intersection, this follows from Example 5.24.The following Theorem 6.14 generalises numerous earlier smoothability results on stretched(by Sally, see [Sal79]), -stretched (by Casnati and Notari, see [CN13]) and almost-stretched (byElias and Valla, see [EV11]) algebras. It is important to understand that, in contrast with thementioned papers, we avoid a full classification of algebras. In the course of the proof we givesome partial classification. Theorem 6.14.
Let A be a local Artin Gorenstein algebra with Hilbert function H A satisfying H A (2) ≤ and H A (3) ≤ . Then A is smoothable.Proof. We proceed by induction on len A , the case len A = 1 being trivial. If A has socle degreethree, then the result follows from Proposition 6.11. Suppose that A has socle degree s ≥ .Let f be a dual socle generator of A in the standard form. If the symmetric decompositionof H A has a term ∆ s − = (0 , q, with q = 0 , then by Example 5.12, we have that A is a limitof algebras of the form B × k , where B satisfies the assumptions H B (2) ≤ and H B (2) ≤ onthe Hilbert function. Then B is smoothable by induction, so also A is smoothable. Further inthe proof we assume that ∆ A,s − = (0 , , .We would like to understand the symmetric decomposition of the Hilbert function H A of A . Since H A satisfies the Macaulay growth condition (see Subsection 2.5) it follows that H A =(1 , n, m, , , . . . , , , . . . , , where the number of “ ” is possibly zero. If follows that the possiblesymmetric decompositions of the Hilbert function are1. (1 , , , . . . , ,
1) + (0 , , , ,
0) + (0 , n − , n − , ,2. (1 , , . . . , ,
1) + (0 , , , . . . , ,
0) + (0 , , , ,
0) + (0 , n − , n − , ,3. (1 , , . . . , ,
1) + (0 , , , ,
0) + (0 , n − , n − , ,4. (1 , . . . ,
1) + (0 , n − , n − , ,5. (1 , , . . . , ,
1) + (0 , n − , n − , ,6. (1 , . . . ,
1) + (0 , , . . . , ,
0) + (0 , n − , n − , ,and that the decomposition is uniquely determined by the Hilbert function. In all cases we have H A (1) ≤ H A (2) ≤ , so f ∈ k [ x , . . . , x ] . Let us analyse the first three cases. In each of themwe have H A (2) = H A (1) + 1 . If H A (1) ≤ , then A is smoothable, see [CN09, Cor 2.4]. Suppose H A (1) ≥ . Since H A (2) ≤ , we have H A (2) = 5 and H A (1) = 4 . In this case the result followsfrom Lemma 6.13 above.It remains to analyse the three remaining cases. The proof is similar to the proof ofLemma 6.13, however here it essentially depends on induction. Let f ≥ be the sum of homoge-neous components of f which have degree at least four. Since f is in the standard form, we have f ≥ ∈ k [ x , x ] . The decomposition of the Hilbert function Apolar ( f ≥ ) is one of the decompo-sitions (1 , . . . , , (1 , . . . , , , (1 , . . . ,
1) + (0 , , . . . , , , depending on the decomposition ofthe Hilbert function of Apolar ( f ) . 29et us fix a vector ˆ h = (1 , , , , . . . , , , , . . . , and take the sets V := n f ∈ k [ x , x ] | H Apolar( f ) = ˆ h o and V := { f ∈ k [ x , . . . , x n ] | f ≥ ∈ V } . By Proposition 4.8 the set V is irreducible and thus V is also irreducible. The Hilbert functionof the apolar algebra of a general member of V is, by Example 6.2, equal to H A . It remains toshow that the apolar algebra of this general member is smoothable.Proposition 4.8 implies that the general member of V has (after a nonlinear change ofcoordinates) the form f + ∂ y f , where f = x s + x s + g for some g of degree at most three.Using Lemma 4.2 we may assume (after another nonlinear change of coordinates) that α y g = 0 .Let B := Apolar ( x s + x s + g ) . We will show that B is smoothable. Since s ≥ · Proposition 5.13 shows that B is limit-reducible. Analysing the fibers of the resulting degener-ation, as in Example 5.15, we see that they have the form B ′ × k , where B ′ = Apolar (cid:16) ˆ f (cid:17) and ˆ f = λ − x s − + x s + g . Then H B ′ (3) = H Apolar ( ˆ f ≥ )(3) ≤ . Moreover, ˆ f ∈ k [ x , . . . , x ] , sothat H B ′ (1) ≤ . Now analysing the possible symmetric decompositions of H B ′ , which are listedabove, we see that H B ′ (2) ≤ H B ′ (1) = 5 . It follows from induction on the length that B ′ issmoothable, thus B ′ × k and B are smoothable. Proposition 6.15.
Let A be a local Artin Gorenstein algebra of socle degree four satisfying len A ≤ . Then A is smoothable.Proof. We proceed by induction on the length of A . Then by Proposition 6.11 (and the fact thatall algebras of socle degree at most two are smoothable) we may assume that all algebras of socledegree at most four and length less than len A are smoothable.If ∆ A, = (0 , q, with q = 0 , then by Example 5.12 the algebra A is a limit of algebras ofthe form A ′ × k , where A ′ has socle degree four. Hence A is smoothable. Therefore we assume q = 0 . Then H A (1) ≤ by Lemma 6.3. Moreover, we may assume H A (1) ≥ since otherwise A is smoothable by [CN09, Cor 2.4].The symmetric decomposition of H A is (1 , n, m, n,
1) + (0 , p, p, for some n, m, p . By thefact that n ≤ and Stanley’s result [Sta96, p. 67] we have n ≤ m , thus n ≤ and H A (2) ≤ H A (1) ≤ . Due to len A ≤ we have four cases: n = 1 , , , and five possible shapes ofHilbert functions: H A = (1 , ∗ , ∗ , , , H A = (1 , ∗ , ∗ , , , H A = (1 , , , , , H A = (1 , , , , , H A = (1 , , , , .The conclusion in the first two cases follows from Theorem 6.14. In the remaining caseswe first look for a suitable irreducible set of dual socle generators parameterising algebras withprescribed H A . We examine the case H A = (1 , , , , . We claim that the set of f ∈ P = k [ x , x , x , x ] in the standard form, which are generators of algebras with Hilbert function H A is irreducible. Since the leading form f of such f has Hilbert function (1 , , , , , the set ofpossible leading forms is irreducible by Proposition 4.9. Then the irreducibility follows fromExample 6.2. The irreducibility in the cases H A = (1 , , , , and H A = (1 , , , , followssimilarly from Proposition 4.10 together with Example 6.2. In the first two cases we see that f is a sum of powers of variables, then Example 5.14 shows that the apolar algebra A of ageneral f is limit-reducible. More precisely, A is limit of algebras of the form A ′ × k , where A ′ has socle degree at most four (compare Example 5.15). Then A is smoothable. In the last caseExample 5.21 gives an unobstructed algebra in this irreducible set. This completes the proof. Lemma 6.16.
Let A be a local Artin Gorenstein algebra with Hilbert function (1 , , , , , .Then A is limit-reducible.Proof. Let s = 5 be the socle degree of A . If ∆ A,s − = (0 , , then A is limit-reducible byExample 5.12, so we assume ∆ A,s − = (0 , , . The only possible symmetric decomposition of30he Hilbert function H A with ∆ A,s − = (0 , , is (1 , , , , ,
1) = (1 , , , , ,
1) + (0 , , , ,
0) + (0 , , , . (12)Let us take a dual socle generator f of A . We assume that f is in the standard form: f = x + f + g , where deg g ≤ . Then H Apolar ( x + f ) = (1 , , , , , . We analyse the possibleHilbert functions of B = Apolar ( f ) . By Lemma 4.2 we may assume that α y f = 0 . Supposefirst that H B (1) ≤ . From (12) it follows that H Apolar ( x + f )(1) = 3 , so that H B (1) = 2 and wemay assume that f ∈ k [ x , x ] . Then by Lemma 4.2 we may further assume α y ( f − x ) = 0 ,then Proposition 5.13 asserts that A = Apolar ( f ) is limit-reducible.Suppose now that H B (1) = 3 . Since x is annihilated by a codimension one space ofquadrics, we have H B (2) ≤ H A (2) + 1 , so there are two possibilities: H B = (1 , , , , or H B = (1 , , , , . By Lemma 6.9 the case H B = (1 , , , , is not possible, so that H B = (1 , , , , . Now by Lemma 6.8 we may consider only the case when (ann S ( f )) is acomplete intersection, then by Lemma 6.7 we have that Apolar (cid:0) x + f (cid:1) is a complete intersec-tion. In this case we will actually prove that A is smoothable.By Example 6.2 the set W of polynomials f with fixed leading polynomial f ≥ and Hilbertfunction H Apolar( f ) = (1 , , , , , is irreducible. Consider the apolar algebra B of the poly-nomial x + f + x x ∈ W . By Proposition 5.10, this algebra is the limit of smoothablealgebras Apolar (cid:0) x + f (cid:1) × Apolar ( x ) , thus it is smoothable. By Corollary 5.20 the algebra B is unobstructed. Thus apolar algebra of every element of W is smoothable; in particular A issmoothable.Now we are ready to prove Theorem 6.17 which is the algebraic counterpart of Theorems Aand B. Theorem 6.17.
Let A be an Artin Gorenstein algebra of length at most . Then either A issmoothable or it is local with Hilbert function (1 , , , . In particular, if A has length at most , then A is smoothable.Proof. By the discussion in Section 2.4 it is enough to consider local algebras. Let A be a localalgebra of length at most and of socle degree s . By H we denote the Hilbert function of A . As mentioned in Subsection 2.4 it is enough to prove A is limit-reducible. On the contrary,suppose that A is strongly non-smoothable in the sense of Definition 2.5. By Example 5.12 wehave ∆ A,s − = (0 , , . Then by Lemma 6.3 we see that either H = (1 , , , or H (1) ≤ . Itis enough to consider H (1) ≤ . If s = 3 then H (2) ≤ H (1) ≤ , so by Proposition 6.11 we mayassume s > . By Proposition 6.15 it follows that we may consider only s ≥ .If H (1) ≤ then A is smoothable by [CN09, Cor 2.4], thus we may assume H (1) ≥ . ByLemma 6.4 we see that H (2) ≤ . Then by Theorem 6.14 we may reduce to the case H (3) ≥ .By Macaulay’s Growth Theorem we have H (2) ≥ . Then P i> H ( i ) ≤ − , so we are leftwith several possibilities: H = (1 , , , , , , , H = (1 , , , , , or H = (1 , ∗ , ∗ , ∗ , , . Inthe first two cases it follows from the symmetric decomposition that ∆ A,s − = (0 , , which isa contradiction. We examine the last case. By Lemma 6.5 there does not exist an algebra withHilbert function (1 , , , , , . Thus the only possibilities are (1 , , , , , , (1 , , , , , and (1 , , , , , . Once more, it can be checked directly that in the first two cases ∆ A,s − = (0 , , .The last case is the content of Lemma 6.16. Remark 6.18.
Assume char k = 0 . In [IE78] Emsalem and Iarrobino analysed the tangentspace to the Hilbert scheme. Iarrobino and Kanev claim that using Macaulay they are able tocheck that the tangent space to H ilb ( P ) has dimension at a point corresponding to a generallocal Gorenstein algebra A with Hilbert function (1 , , , , see [IK99, Lem 6.21], see also [CN11]for further details. Since < (1 + 6 + 6 + 1) · this shows that A is non-smoothable. Moreover,since all algebras of degree at most are smoothable, A is strongly non-smoothable.
31o prove Theorem B, we need to show that the non-smoothable part of H ilb G P n (for n ≥ ) isirreducible. The algebraic version of (a generalisation of) this statement is the following lemma. Lemma 6.19.
Let n ≥ m be natural numbers and V ⊆ P ≤ = k [ x , . . . , x n ] ≤ be the set of f ∈ P such that H Apolar( f ) = (1 , m, m, . Then V is constructible and irreducible.Proof. Let V gr = V ∩ P denote the set of graded algebras with Hilbert function (1 , m, m, .This is a constructible subset of P . To an element f ∈ V gr we may associate the tangent spaceto Apolar ( f ) , which is isomorphic to S y f . We define { ( f , [ W ]) ∈ V gr × Gr( m, n ) | W ⊇ S y f } , which is an open subset in a vector bundle { ( f , [ W ]) ∈ P × Gr( m, n ) | W ⊇ S y f } over Gr( m, n ) , given by the condition dim S y f ≥ m . Let f ∈ V and write it as f = f + f ≤ , where deg f ≤ ≤ . Then H Apolar( f ) = (1 , m, m, . Therefore we obtain a morphism ϕ : V → V gr sending f to f . We will analyse its fibers. Let f ∈ V gr and f = f + f ≤ ∈ P ≤ , where deg f ≤ ≤ . Then H Apolar( f ) = (1 , M, m, for some M ≥ m . Moreover M = m if and only if α y f ≤ is a partial of f for every α annihilating f . The fiber of ϕ over f is an affine subspaceof P ≤ defined by these conditions and the morphism { ( f = f + f ≤ , [ W ]) ∈ V × Gr( m, n ) | W ⊇ S y f } → { ( f , [ W ]) ∈ V gr × Gr( m, n ) | W ⊇ S y f } is a projection from a vector bundle, which is thus irreducible. Since V admits a surjection fromthis bundle, it is irreducible as well. Moreover, the above shows that V is constructible. Proof of Theorems A and B.
The locus of points of the Hilbert scheme corresponding to smooth(i.e. reduced) algebras of length d is irreducible, as an image of an open subset of the d –symmetricproduct of P n , and smooth. The locus of points corresponding to smoothable algebras is theclosure of the aforementioned locus, so it is also irreducible. If d ≤ or d ≤ and n ≤ , thislocus is the whole Hilbert scheme by Theorem 6.17 and the claim follows.Now consider the case d = 14 and n ≥ . Let V be the set of points of the Hilbert schemecorresponding to local Gorenstein algebras with Hilbert function (1 , , , . By Remark 6.18these are the only non-smoothable algebras of length , thus they deform only to local algebraswith the same Hilbert function. Therefore, V is a sum of irreducible components of the Hilbertscheme. We will prove that V is an irreducible set, whose general point is smooth.Let V p ⊆ V denote the set consisting of schemes supported at a fixed point p ∈ P n . Then V is dominated by a set V p × P n . Note that an irreducible scheme supported at a point p may be identified with a Gorenstein quotient of the power series ring having Hilbert function (1 , , , . These quotients are parameterised by the dual generators. More precisely, the set of V of f ∈ k [ x , . . . , x n ] ≤ such that H Apolar( f ) = (1 , , , gives a morphism V → V p ⊆ H ilb G P n which sends f to Spec Apolar ( f ) supported at p (see subsection 6.1). Since V → V p is surjectiveand V is irreducible by Lemma 6.19, we see that V p is irreducible. Then V is irreducible as well.Take a smooth point of H ilb G P which corresponds to an algebra A with Hilbert function (1 , , , . Then any point of H ilb G P n corresponding to an embedding Spec A ⊆ P n is smoothby [CN09, Lem 2.3]. This concludes the proof. We wish to express our thanks to A.A. Iarrobino and P.M. Marques for inspiring conversations.Moreover we are also sincerely grateful to W. Buczyńska and J. Buczyński for their care, support32nd hospitality during the preparation of this paper. We also thank J. Buczyński for explainingthe proof of Proposition 4.7. We thank the referee for careful reading and suggesting a numberof improvements. The examples were obtained with the help of Magma computing software,see [BCP97].
References [BB14] W. Buczyńska and J. Buczyński. Secant varieties to high degree Veronese reem-beddings, catalecticant matrices and smoothable Gorenstein schemes.
J. AlgebraicGeom. , 23:63–90, 2014.[BCP97] W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system. I. The userlanguage.
J. Symbolic Comput. , 24(3-4):235–265, 1997. Computational algebra andnumber theory (London, 1993).[BCR12] C. Bertone, F. Cioffi, and M. Roggero. A division algorithm in an affine frameworkfor flat families covering Hilbert schemes. arXiv preprint arXiv:1211.7264 , 2012.[BE77] D. A. Buchsbaum and D. Eisenbud. Algebra structures for finite free resolutions, andsome structure theorems for ideals of codimension . Amer. J. Math. , 99(3):447–485,1977.[BGI11] A. Bernardi, A. Gimigliano, and M. Idà. Computing symmetric rank for symmetrictensors.
J. Symbolic Comput. , 46(1):34–53, 2011.[BH93] W. Bruns and J. Herzog.
Cohen-Macaulay rings , volume 39 of
Cambridge Studies inAdvanced Mathematics . Cambridge University Press, Cambridge, 1993.[BJ] J. Buczyński and J. Jelisiejew. On smoothability. Preprint available at [BJJM] J. Buczyński, T. Januszkiewicz, J. Jelisiejew, and M. Michałek. On the existence of k -regular maps. In preparation.[CEVV09] D. A. Cartwright, D. Erman, M. Velasco, and B. Viray. Hilbert schemes of 8 points. Algebra Number Theory , 3(7):763–795, 2009.[CJN13] G. Casnati, J. Jelisiejew, and R. Notari. On the rationality of Poincarè series ofGorenstein algebras via Macaulay’s correspondence.
ArXiv e-prints , July 2013.[CN09] G. Casnati and R. Notari. On the Gorenstein locus of some punctual Hilbert schemes.
J. Pure Appl. Algebra , 213(11):2055–2074, 2009.[CN11] G. Casnati and R. Notari. On the irreducibility and the singularities of the Gorensteinlocus of the punctual Hilbert scheme of degree 10.
J. Pure Appl. Algebra , 215(6):1243–1254, 2011.[CN13] G. Casnati and R. Notari. A structure theorem for 2-stretched Gorenstein algebras. arXiv preprint arXiv:1312.2191 , 2013.[CN14] G. Casnati and R. Notari. On the Gorenstein locus of the punctual Hilbert schemeof degree 11.
J. Pure Appl. Algebra , 218(9):1635–1651, 2014.[Eis95] D. Eisenbud.
Commutative algebra , volume 150 of
Graduate Texts in Mathematics .Springer-Verlag, New York, 1995. With a view toward algebraic geometry.33ER15] J. Elias and M. E. Rossi. Analytic isomorphisms of compressed local algebras.
Proc.Amer. Math. Soc. , 143(3):973–987, 2015.[Ell75] G. Ellingsrud. Sur le schéma de Hilbert des variétés de codimension dans P e à cônede Cohen-Macaulay. Ann. Sci. École Norm. Sup. (4) , 8(4):423–431, 1975.[Ems78] J. Emsalem. Géométrie des points épais.
Bull. Soc. Math. France , 106(4):399–416,1978.[ER12] J. Elias and M. E. Rossi. Isomorphism classes of short Gorenstein local rings viaMacaulay’s inverse system.
Trans. Amer. Math. Soc. , 364(9):4589–4604, 2012.[EV11] J. Elias and G. Valla. Isomorphism classes of certain Artinian Gorenstein algebras.
Algebr. Represent. Theory , 14(3):429–448, 2011.[Fog68] J. Fogarty. Algebraic families on an algebraic surface.
Amer. J. Math , 90:511–521,1968.[Ger99] A. V. Geramita. Catalecticant varieties. In
Commutative algebra and algebraic geom-etry (Ferrara) , volume 206 of
Lecture Notes in Pure and Appl. Math. , pages 143–156.Dekker, New York, 1999.[Gro95] A. Grothendieck. Techniques de construction et théorèmes d’existence en géométriealgébrique. IV. Les schémas de Hilbert. In
Séminaire Bourbaki, Vol. 6 , pages Exp.No. 221, 249–276. Soc. Math. France, Paris, 1995.[Har66] R. Hartshorne. Connectedness of the Hilbert scheme.
Publications Mathématiques del’IHÉS , 29(1):7–48, 1966.[Har77] R. Hartshorne.
Algebraic geometry . Springer-Verlag, New York, 1977. GraduateTexts in Mathematics, No. 52.[Har10] R. Hartshorne.
Deformation theory , volume 257 of
Graduate Texts in Mathematics .Springer, New York, 2010.[Iar72] A. A. Iarrobino. Reducibility of the families of -dimensional schemes on a variety. Invent. Math. , 15:72–77, 1972.[Iar77] A. A. Iarrobino. Punctual Hilbert schemes.
Mem. Amer. Math. Soc. ,10(188):viii+112, 1977.[Iar84] A. A. Iarrobino. Compressed algebras: Artin algebras having given socle degrees andmaximal length.
Trans. Amer. Math. Soc. , 285(1):337–378, 1984.[Iar94] A. A. Iarrobino. Associated graded algebra of a Gorenstein Artin algebra.
Mem.Amer. Math. Soc. , 107(514):viii+115, 1994.[IE78] A. A. Iarrobino and J. Emsalem. Some zero-dimensional generic singularities; finitealgebras having small tangent space.
Compositio Math. , 36(2):145–188, 1978.[IK99] A. A. Iarrobino and V. Kanev.
Power sums, Gorenstein algebras, and determinantalloci , volume 1721 of
Lecture Notes in Mathematics . Springer-Verlag, Berlin, 1999.Appendix C by Iarrobino and Steven L. Kleiman.[Jel13] J. Jelisiejew. Deformations of zero-dimensional schemes and applications.
ArXive-print arXiv:1307.8108 , 2013. 34Jel14] Joachim Jelisiejew. Local finite-dimensional Gorenstein k-algebras having Hilbertfunction (1,5,5,1) are smoothable.
J. Algebra Appl. , 13(8):1450056 (7 pages), 2014.[KMR98] J. O. Kleppe and R. M. Miró-Roig. The dimension of the Hilbert scheme of Gorensteincodimension subschemes. J. Pure Appl. Algebra , 127(1):73–82, 1998.[Kun05] E. Kunz.
Introduction to plane algebraic curves . Birkhäuser Boston, Inc., Boston,MA, 2005. Translated from the 1991 German edition by Richard G. Belshoff.[LO13] J. M. Landsberg and G. Ottaviani. Equations for secant varieties of Veronese andother varieties.
Ann. Mat. Pura Appl. (4) , 192(4):569–606, 2013.[Mat86] H. Matsumura.
Commutative ring theory , volume 8 of
Cambridge Studies in AdvancedMathematics . Cambridge University Press, Cambridge, 1986. Translated from theJapanese by M. Reid.[MR92] R. M. Miró-Roig. Nonobstructedness of Gorenstein subschemes of codimension in P n . Beiträge Algebra Geom. , (33):131–138, 1992.[Rai12] C. Raicu. Secant varieties of Segre-Veronese varieties.
Algebra Number Theory ,6(8):1817–1868, 2012.[Sal79] J. D. Sally. Stretched Gorenstein rings.
J. London Math. Soc. (2) , 20(1):19–26, 1979.[Sta96] R. P. Stanley.
Combinatorics and commutative algebra , volume 41 of
Progress inMathematics . Birkhäuser Boston Inc., Boston, MA, second edition, 1996.[VV78] P. Valabrega and G. Valla. Form rings and regular sequences.
Nagoya Math. J. ,72:93–101, 1978.Gianfranco Casnati,Dipartimento di Scienze Matematiche, Politecnico di Torino,corso Duca degli Abruzzi 24, 10129 Torino, Italye-mail: [email protected]
Joachim Jelisiejew,Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw,Banacha 2, 02-097 Warsaw, Poland [email protected]
Roberto Notari,Dipartimento di Matematica, Politecnico di Milano,via Bonardi 9, 20133 Milano, Italye-mail: [email protected]@polimi.it