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Let $I$ be a homogeneous ideal of $\Bbbk[x_0,\ldots,x_n]$. To compare $I^{(m)}$, the $m$-th symbolic power of $I$, with $I^m$, the regular $m$-th power, we introduce the $m$-th symbolic defect of $I$, denoted $\operatorname{sdefect}(I,m)$. Precisely, $\operatorname{sdefect}(I,m)$ is the minimal number of generators of the $R$-module $I^{(m)}/I^m$, or equivalently, the minimal number of generators one must add to $I^m$ to make $I^{(m)}$. In this paper, we take the first step towards understanding the symbolic defect by considering the case that $I$ is either the defining ideal of a star configuration or the ideal associated to a finite set of points in $\mathbb{P}^2$. We are specifically interested in identifying ideals $I$ with $\operatorname{sdefect}(I,2) = 1$., To appear in Journal of Pure and Applied Algebra; revised at referees' suggestion. Fixed typos and clarified writing, included additional references, shortened proof of Thm 6.3

Given the space V = P ( d+n 1 n 1 ) 1 of forms of degree d in n variables, and given an integer l > 1 and a partition λ of d = d1 + ··· + dr, it is in general an open problem to obtain the dimensions of the l-secant varieties σl(Xn 1,λ) for the subvariety Xn 1,λ ⊂ V of hypersurfaces whose defining forms have a factorization into forms of degrees d1,...,dr. Modifying a method from intersection theory, we relate this problem to the study of the Weak Lefschetz Property for a class of graded algebras, based on which we give a conjectural formula for the dimension of σl(Xn 1,λ) for any choice of parameters n,l and λ. This conjecture gives a unifying framework subsuming all known results. Moreover, we unconditionally prove the formula in many cases, considerably extending previous results, as a consequence of which we verify many special cases of previously posed conjectures for dimensions of secant varieties of Segre varieties. In the special case of a partition with two parts (i.e., r = 2), we also relate this problem to a conjecture by Froberg on the Hilbert function of an ideal generated by general forms.

We consider ideals in a polynomial ring that are generated by regular sequences of homogeneous polynomials and are stable under the action of the symmetric group permuting the variables. In previous work, we determined the possible isomorphism types for these ideals. Following up on that work, we now analyze the possible degrees of the elements in such regular sequences. For each case of our classification, we provide some criteria guaranteeing the existence of regular sequences in certain degrees., Comment: 22 pages

Star configurations are certain unions of linear subspaces of projective space that have been studied extensively. We develop a framework for studying a substantial generalization, which we call matroid configurations, whose ideals generalize Stanley-Reisner ideals of matroids. Such a matroid configuration is a union of complete intersections of a fixed codimension. Relating these to the Stanley-Reisner ideals of matroids and using methods of Liaison Theory allows us, in particular, to describe the Hilbert function and minimal generators of the ideal of, what we call, a hypersurface configuration. We also establish that the symbolic powers of the ideal of any matroid configuration are Cohen-Macaulay. As applications, we study ideals coming from certain complete hypergraphs and ideals derived from tetrahedral curves. We also consider Waldschmidt constants and resurgences. In particular, we determine the resurgence of any star configuration and many hypersurface configurations. Previously, the only non-trivial cases for which the resurgence was known were certain monomial ideals and ideals of finite sets of points. Finally, we point out a connection to secant varieties of varieties of reducible forms., Comment: 16 pages

We describe the graded characters and Hilbert functions of certain graded artinian Gorenstein quotients of the polynomial ring which are also representations of the symmetric group. Specifically, we look at those algebras whose socles are trivial representations and whose principal apolar submodules are generated by the sum of the orbit of a power of a linear form.

In this paper we introduce a new method to produce lower bounds for the Waring rank of symmetric tensors. We also introduce the notion of $e$-computability and we use it to prove that Strassen's Conjecture holds in infinitely many new cases., arXiv admin note: text overlap with arXiv:1412.2975

Let $\lambda =[d_1,\dots,d_r]$ be a partition of $d$. Consider the variety $\mathbb{X}_{2,\lambda} \subset \mathbb{P}^N$, $N={d+2 \choose 2}-1$, parameterizing forms $F\in k[x_0,x_1,x_2]_d$ which are the product of $r\geq 2$ forms $F_1,\dots,F_r$, with deg$F_i = d_i$. We study the secant line variety $\sigma_2(\mathbb{X}_{2,\lambda})$, and we determine, for all $r$ and $d$, whether or not such a secant variety is defective. Defectivity occurs in infinitely many "unbalanced" cases., Comment: 19 pages, 3 figures. In new version Typos corrected, exposition improved

After showing that the General Cayley–Bacharach Conjecture formulated by D. Eisenbud, M. Green, and J. Harris (1996) [6] is equivalent to a conjecture about the region of uniformity of a zero-dimensional complete intersection, we prove this conjecture in a number of special cases. In particular, after splitting the conjecture into several intervals, we prove it for the first, the last and part of the penultimate interval. Moreover, we generalize the uniformity results of J. Hansen (2003) [12] and L. Gold, J. Little, and H. Schenck (2005) [9] to level schemes and apply them to obtain bounds for the minimal distance of generalized Reed–Muller codes.

Star configurations are certain unions of linear subspaces of projective space. They have appeared in several different contexts: the study of extremal Hilbert functions for fat point schemes in the plane; the study of secant varieties of some classical algebraic varieties; the study of the resurgence of projective schemes. In this paper we study some algebraic properties of the ideals defining star configurations, including getting partial results about Hilbert functions, generators and minimal free resolutions of the ideals and their symbolic powers. We also show that their symbolic powers define arithmetically Cohen–Macaulay subschemes and we obtain results about the primary decompositions of the powers of the ideals. As an application, we compute the resurgence for the ideal of the codimension n − 1 star configuration in P n in the monomial case (i.e., when the number of hyperplanes is n + 1 ).

We consider complete intersection ideals in a polynomial ring over a field of characteristic zero that are stable under the action of the symmetric group permuting the variables. We determine the possible representation types for these ideals, and describe formulas for the graded characters of the corresponding quotient rings., 14 pages, includes referee's suggestions for publication in Communications in Algebra

Polito SFX(opens in a new window)| Export | Download | Add to List | More... Annali della Scuola normale superiore di Pisa - Classe di scienze Volume 15, 2016, Pages 69-84 On the Hilbert function of lines union one non-reduced point (Article) Carlini, E.a , Catalisano, M.V.bc , Geramita, A.V.d a Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca Degli Abbruzzi 24, Torino, Italy b DIME Dipartimento di Ingegneria Meccanica Energetica Gestionale E Dei Trasporti, Universita Degli Studi di Genova, Piazzale Kennedy pad. D, Genova, Italy c Department of Mathematics and Statistics, Queen's University, Kingston, ON, Canada View additional affiliations View references (18) Abstract In this paper we consider the problem of determining the Hilbert function of schemes XcP which are the generic union of s lines and one m- multiple point. We completely solve this problem for any s and m when n ≥ 4. When n = 3 we find several defective such schemes and conjecture that they are the only ones. We verify this conjecture in several cases.

A configuration of linear spaces in a projective space is a finite collection of linear subspaces. In this paper we study the degree 2 part of the ideal of such objects. More precisely, for a generic configuration of linear spaces Λ we determine HF(Λ,2), i.e. the Hilbert function of Λ in degree 2.

The bigraded Hilbert function and the minimal free resolutions for the diagonal coinvariants of the dihedral groups are exhibited, as well as for all their bigraded invariant Gorenstein quotients.

Let X ⊂ P n be a closed subscheme and let HF ( X , ⋅ ) and hp ( X , ⋅ ) denote, respectively, the Hilbert function and the Hilbert polynomial of X. We say that X has bipolynomial Hilbert function if HF ( X , d ) = min { hp ( P n , d ) , hp ( X , d ) } for every d ∈ N . We show that if X consists of a plane and generic lines, then X has bipolynomial Hilbert function. We also conjecture that generic configurations of non-intersecting linear spaces have bipolynomial Hilbert function.

The paper [10] raised the question of what the possible Hilbert functions are for fat point subschemes of the form 2p 1+...+2p r, for all possible choices ofr distinct points in ℙ2. We study this problem forr points in ℙ2 over an algebraically closed fieldk of arbitrary characteristic in case eitherr ≤ 8 or the points lie on a (possibly reducible) conic. In either case, it follows from [17, 18] that there are only finitely many configuration types of points, where our notion of configuration type is a generalization of the notion of a representable combinatorial geometry, also known as a representable simple matroid. (We sayp 1, ...,p r andp 1 ′, ... ,p r ′ have the sameconfiguration type if for all choices of nonnegative integersm i ,Z=m 1 p 1+...+m r p r andZ′=m 1 p 1 ′+...+m r p r ′ have the same Hilbert function.) Assuming either that 7 ≤r ≤ 8 (see [12] for the casesr ≤6) or that the pointsp i lie on a conic, we explicitly determine all the configuration types, and show how the configuration type and the coefficientsm i determine (in an explicitly computable way) the Hilbert function (and sometimes the graded Betti numbers) ofZ=m 1 p 1+...+m r p r . We demonstrate our results by explicitly listing all Hilbert functions for schemes ofr≤ 8 double points, and for each Hilbert function we state precisely how the points must be arranged (in terms of the configuration type) to obtain that Hilbert function.

In this paper we investigate some algebraic and geometric consequences which arise from an extremal bound on the Hilbert function of the general hyperplane section of a variety (Green's Hyperplane Restriction Theorem). These geometric consequences improve some results in this direction first given by Green and extend others by Bigatti, Geramita, and Migliore. Other applications of our detailed investigation of how the Hilbert polynomial is written as a sum of binomials, are to conditions that must be satisfied by a polynomial if it is to be the Hilbert polynomial of a non-degenerate integral subscheme of P n (a problem posed by R.P. Stanley). We also give some new restrictions on the Hilbert function of a zero-dimensional reduced scheme with the Uniform Position Property.

Let \(A= (a_{ij})\) be a non-negative integer \(k\times k\) matrix. \(A\) is a homogeneous matrix if \(a_{ij} + a_{kl}=a_{il} + a_{kj}\) for any choice of the four indexes. We ask: If \(A\) is a homogeneous matrix and if \(F\) is a form in \(\mathbb {C}[x_1, \dots x_n]\) with \(deg(F) = \mathrm{trace}(A)\), what is the least integer, \(s(A)\), so that \(F = detM_1 + \cdots + detM_{s(A)}\), where the \(M_i = (F^i_{lm})\) are \(k\times k\) matrices of forms and \(deg F^i_{lm} = a_{lm}\) for every \(1\le i \le s(A)\)? We consider this problem for \(n\ge 4\) and we prove that \(s(A) \le k^{n-3}\) and \(s(A)

Let $F$ be a homogeneous form of degree $d$ in $n$ variables. A Waring decomposition of $F$ is a way to express $F$ as a sum of $d^{th}$ powers of linear forms. In this paper we consider the decompositions of a form as a sum of expressions, each of which is a fixed monomial evaluated at linear forms., Comment: 12 pages; Section 5 added in this version

Let V t = P 1 × ⋯ × P 1 ( t -copies) embedded in P N ( N = 2 t - 1 ) via the Segre embedding. Let ( V t ) s be the subvariety of P N which is the closure of the union of all the secant P s - 1 's to V t . The expected dimension of ( V t ) s is min { st + ( s - 1 ) , N } . This is not the case for ( V 4 ) 3 , which we conjecture is the only defective example in this infinite family. We prove (Theorem 2.3): if e t = [ 2 t t + 1 ] ≡ δ t ( mod 2 ) and s t = e t - δ t then ( V t ) s has the expected dimension, except possibly when s = s t + 1 . Moreover, whenever t = 2 k - 1 , ( V t ) s has the expected dimension for every s .

In this paper we prove the existence of minimal level artinian graded algebras having socle degree r and type t and describe their h-vector in terms of the r-binomial expansion of t. We also investigate the graded Betti numbers of such algebras and completely describe their extremal resolutions. We also show that any set of points in ℙ n whose Hilbert function has first difference as described above, must satisfy the Cayley-Bacharach property.

We prove that a general polynomial form of degree d in 4 variables, over the complex field, can be written as the sum of two determinants of 2 × 2 matrices of forms, with given degree matrix (a ij ), for any choice of non-negative integers a ij ≤ d with a 11 + a 22 = a 12 + a 21 = d.

We study the dimensions of the higher secant varieties to the tangent varieties of Veronese varieties. Our approach, generalizing that of Terracini, concerns 0-dimensional schemes which are the union of second infinitesimal neighbourhoods of generic points, each intersected with a generic double line. We find the deficient secant line varieties for all the Veroneseans and all the deficient higher secant varieties for the quadratic Veroneseans. We conjecture that these are the only deficient secant varieties in this family and prove this up to secant projective 4-spaces.

Let H be the Hilbert function of some set of distinct points in ℙn and let α = α(H) be the least degree of a hypersurface of ℙn containing these points. Write α = ds + ds–1 + … + d1 (where di > 0). We canonically decompose H into s other Hilbert functions and show how to find sets of distinct points , lying on reduced hypersurfaces of degrees ds, …, d1 (respectively) such that the Hilbert function of is and the Hilbert function of is H. Some extremal properties of this canonical decomposition are also explored.

The Hilbert function of a homogeneous ideal in R=k[x0 , ..., xn], k a field, is a much studied object. This is not surprising since the Hilbert function encodes important algebraic, combinatorial, and geometric information about the ideal. The fact that recent computer algebra developments have made the Hilbert function computable has not only sustained interest in them but sparked interest in many new questions about them. In this paper, we will concentrate on the Hilbert functions which are the Hilbert functions of points in P. From [11], we know that this is the same as studying 0-dimensional differentiable O-sequences (equivalently, the Hilbert functions of graded artinian quotients of k[x1 , ..., xn]). In our earlier paper [9], we began a discussion of n-type vectors and showed that they were in 1 1 correspondence with Hilbert functions of doi:10.1006 aima.1998.1889, available online at http: www.idealibrary.com on

Given a finite setX, of points in projective space for which the Hilbert function is known, a standard result says that there exists a subset of this finite set whose Hilbert function is"as big as possible"inside X. Given a finite set of points in projective space for which the minimal free resolution of its homogeneous ideal is known, what can be said about possible resolutions of ideals of subsets of this finite set? We first give a maximal rank type description of the most generic possible resolution of a subset. Then we show, via two very different kinds of counterexamples, that this generic resolution is not always achieved. However, we show that it is achieved for sets of points in projective two space: given any finite set of points in projective two space for which the minimal free resolution is known, there must exist a subset having the predicted resolution.

We study the Hilbert polynomials of finitely generated graded algebras R, with generators not all of degree one (i.e. non-standard). Given an expression P(R,t)=a(t)/(1-tl ) n for the Poincare series of R as a rational function, we study for 0 ≤ i ≤ l the graded subspaces ⊗ kRkl+i (which we denote R[l;i]) of R, in particular their Poincare series and Hilbert functions. We prove, for example, that if R is Cohen-Macaulay then the Hilbert polynomials of all non-zeroR[l;i] share a common degree. Furthermore, if R is also a domain then these Hilbert polynomials have the same leading coefficient.

This paper contains two essentially independent results in the invariant theory of finite groups. First we prove that, for any faithful representation of a non-trivial p-group over a field of characteristic p, the ring of vector invariants ofmcopies of that representation is not Cohen-Macaulay for m ≥ 3. In the second section of the paper we use Poincaré series methods to produce upper bounds for the degrees of the generators for the ring of invariants as long as that ring is Gorenstein. We prove that, for a finite non-trivial group G and a faithful representation of dimension n with n > 1, if the ring of invariants is Gorenstein then the ring is generated in degrees less than or equal to n(|G| − 1). If the ring of invariants is a hypersurface, the upper bound can be improved to |G|.

We prove that allk-configurations of points in P 3have the extremal resolution for a Cohen–Macaulay scheme with the Hilbert function.

We explore the connection between ideals of fat points (which correspond to subschemes of P nobtained by intersecting (mixed) powers of ideals of points), and piecewise polynomial functions (splines) on ad-dimensional simplicial complex Δ embedded inRd. Using the inverse system approach introduced by Macaulay [ 11 ], we give a complete characterization of the free resolutions possible for ideals ink[x, y] generated by powers of homogeneous linear forms (we allow the powers to differ). We show how ideals generated by powers of homogeneous linear forms are related to the question of determining, for some fixed Δ, the dimension of the vector space of splines on Δ of degree less than or equal tok. We use this relationship and the results above to derive a formula which gives the number of planar (mixed) splines in sufficiently high degree.

We give a construction for Gorenstein ideals of codimension 4 which we believe have maximal graded Betti numbers for their Hilbert function.

In this paper we investigate some relationships between codimension 2 Cohen-Macaulay graded rings and codimension three Artinian Gorenstein graded rings and use these to make some comments about the classifying spaces of such Gorenstein algebras having a fixed Hubert function.

It is known, from work of Diesel, which graded Betti numbers are possible for Artinian Gorenstein height three ideals. In this paper we show that any such set of graded Betti numbers in fact occurs for a reduced set of points in P 3 \mathbb P^3 , a stick figure in P 4 \mathbb P^4 , or more generally, a good linear configuration in P n \mathbb P^n . Consequently, any Gorenstein codimension three scheme specializes to such a “nice” configuration, preserving the graded Betti numbers in the process. This is the codimension three Gorenstein analog of a classical result of arithmetically Cohen-Macaulay codimension two schemes.

F. S. Macaulay gave necessary and sufficient conditions on the growth of a nonnegative integer-valued function which determine when such a function can be the Hilbert function of a standard graded k k -algebra. We investigate some algebraic and geometric consequences which arise from the extremal cases of Macaulay’s theorem. Our work also builds on the fundamental work of G. Gotzmann. Our principal applications are to the study of Hilbert functions of zero-schemes with uniformity conditions. As a consequence, we have new strong limitations on the possible Hilbert functions of the points which arise as a general hyperplane section of an irreducible curve.

Let V n V_n be the Segre embedding of P 1 × ⋯ × P 1 {\mathbb {P}^1}\times \cdots \times {\mathbb {P}^1} ( n n times). We prove that the higher secant varieties σ s ( V n ) \sigma _s(V_n) always have the expected dimension, except for σ 3 ( V 4 ) \sigma _3(V_4) , which is of dimension 1 less than expected.

A set of s s points in P d {\mathbb {P}^d} is called a Cayley-Bacharach scheme ( CB {\text {CB}} -scheme), if every subset of s − 1 s - 1 points has the same Hilbert function. We investigate the consequences of this "weak uniformity." The main result characterizes CB {\text {CB}} -schemes in terms of the structure of the canonical module of their projective coordinate ring. From this we get that the Hilbert function of a CB {\text {CB}} -scheme X X has to satisfy growth conditions which are only slightly weaker than the ones given by Harris and Eisenbud for points with the uniform position property. We also characterize CB {\text {CB}} -schemes in terms of the conductor of the projective coordinate ring in its integral closure and in terms of the forms of minimal degree passing through a linked set of points. Applications include efficient algorithms for checking whether a given set of points is a CB {\text {CB}} -scheme, results about generic hyperplane sections of arithmetically Cohen-Macaulay curves and inequalities for the Hilbert functions of Cohen-Macaulay domains.

We generalize an example, due to Sylvester, and prove that any monomial of degree $d$ in $\mathbb R[x_0, x_1]$, which is not a power of a variable, cannot be written as a linear combination of fewer than $d$ powers of linear forms., 5 pages

Si considera la seguente domanda: quando punti intersezioni completa di tipo (a,b,c) si trovano sulla generica superficie di grado d di P3? Diamo una risposta asintotica per ogni scelta si a,b,c e d alla questione dell'esistenza. Una risposta completa e' fornita per valori piccoli di a,b,e c

We ask when certain complete intersections of codimension $r$ can lie on a generic hypersurface in $\PP^n$. We give a complete answer to this question when $2r \leq n+2$ in terms of the degrees of the hypersurfaces and of the degrees of the generators of the complete intersection.

We determine for all values of (a,b,c), the dimension of the secant varieties of the (a,b,c)-embedding of P1xP1xP1 and classify all the defective ones.

If $\X \subset ��^n$ is a reduced and irreducible projective variety, it is interesting to find the equations describing the (higher) secant varieties of $\X$. In this paper we find those equations in the following cases: $\X = ��^{n_1}\times...\times��^{n_t}\times��^n$ is the Segre embedding of the product and $n$ is "large" with respect to the $n_i$ (Theorem 2.4); $\X$ is a Segre-Veronese embedding of some products with 2 or three factors; $\X$ is a Del Pezzo surface., 17 pages, minor changes for section 3 and references