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A. V. Geramita

Publications and source records attributed to A. V. Geramita.

16 recordsLinked to original sources
arXiv2021

Secant Varieties of the Varieties of Reducible Hypersurfaces in ${\mathbb P}^n$

Given the space $V={\mathbb P}^{\binom{d+n-1}{n-1}-1}$ of forms of degree $d$ in $n$ variables, and given an integer $\ell>1$ and a partition $λ$ of $d=d_1+\cdots+d_r$, it is in general an open problem to obtain the dimensions of the $\ell$-secant varieties $σ_\ell ({\mathbb X}_{n-1,λ})$ for the subvariety ${\mathbb X}_{n-1,λ} \subset V$ of hypersurfaces whose defining forms have a factorization into forms of degrees $d_1,\ldots,d_r$. 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 $σ_\ell({\mathbb X}_{n-1,λ})$ for any choice of parameters $n,\ell$ 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 Fröberg on the Hilbert function of an ideal generated by general forms.

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arXiv2015

Matroid configurations and symbolic powers of their ideals

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.

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arXiv2012

Star configurations in $\mathbb P^n$

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 $\pr{n}$ in the monomial case (i.e., when the number of hyperplanes is $n+1$).

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arXiv2011

Hilbert functions of double point schemes in P^2

We study the question of whether there is a minimum Hilbert functions for double point schemes whose support is $s$ points with the generic Hilbert functions. Previous work shows that the question has an affirmative answer for $s \le 9$ and for $s$ equal to a binomial coefficient number. In this paper, we give evidence in the case when $s$ equals to a binomial coefficient number plus 1, and give an affirmative answer to the question when $s = 11$.

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arXiv2011

Hilbert functions of fat point subschemes of the plane: the two-fold way

Two approaches for determining Hilbert functions of fat point subschemes of $\mathbb P^2$ are demonstrated. A complete determination of the Hilbert functions which occur for 9 double points is given using the first approach, extending results obtained in a previous paper using the second approach. In addition the second approach is used to obtain a complete determination of the Hilbert functions for $n\geq 9$ $m$-multiple points for every $m$ if the points are smooth points of an irreducible plane cubic curve. Additional results are obtained using the first approach for $n\geq 9$ double points when the points lie on an irreducible cubic (but now are not assumed to be smooth points of the cubic).

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arXiv2010

Classifying Hilbert functions of fat point subschemes in $\mathbb P^2$

A recent paper by the first and third authors together with Sabourin 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 of $r$ distinct points in the projective plane. We study this problem for $r$ points in the plane over an algebraically closed field $k$ of arbitrary characteristic in case either $r \le 8$ or the points lie on a (possibly reducible) conic. In either case, it follows from work of the second author 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 say $p_1,...,p_r$ and $p'_1,...,p'_r$ have the same {\it configuration type} if for all choices of nonnegative integers $m_i$, $Z=m_1p_1+...+m_rp_r$ and $Z'=m_1p'_1+...+m_rp'_r$ have the same Hilbert function.) Assuming either that $7 \le r\le 8$ (see recent work of Guardo and the second author for the cases $r\le 6$) or that the points $p_i$ lie on a conic, we explicitly determine all the configuration types, and show how the configuration type and the coefficients $m_i$ determine (in an explicitly computable way) the Hilbert function (and sometimes the graded Betti numbers) of $Z=m_1p_1+...+m_rp_r$. We demonstrate our results by explicitly listing all Hilbert functions for schemes of $r\le 8$ 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.

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arXiv2009

Bipolynomial Hilbert functions

Let X 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 non-negative integer d. 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.

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arXiv2009

Subspace arrangements, configurations of linear spaces and the quadrics containing them

A subspace arrangement in a vector space is a finite collection of vector subspaces. Similarly, 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 L we determine HF(L,2), i.e. the Hilbert function of L in degree 2.

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arXiv2008

Complete intersection points on general surfaces in $\PP^3$

In this paper we consider the existence of complete intersection points of type $(a,b,c)$, on the generic degree $d$ surface of $\PP^3$. For any choice of $a, b, c$ we resolve the existence question asymptotically, i.e. for all $d \gg 0$. For small values of $a, b, c$ we resolve the existence problem completely.

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arXiv2008

Complete intersections on general hypersurfaces

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.

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arXiv2006

On the ideals of Secant Varieties to certain rational varieties

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.

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arXiv2004

On the first infinitesimal neighborhood of a linear configuration of points in $\mathbb P^2$

We consider the following open questions. Fix a Hilbert function, $h$, that occurs for a reduced zero-dimensional subscheme of $\mathbb P^2$. Among all subschemes, $X$, with Hilbert function $h$, what are the possible Hilbert functions and graded Betti numbers for the first infinitesimal neighborhood, $Z$, of $X$ (i.e. the double point scheme supported on $X$)? Is there a minimum ($h^{\min}$) and maximum ($h^{\max}$) such function? The numerical information encoded in $h$ translates to a {\it type vector}, which allows us to find unions of points on lines, called {\it linear configurations}, with Hilbert function $h$. We give necessary and sufficient conditions for the Hilbert function and graded Betti numbers of the first infinitesimal neighborhoods of {\it all} such linear configurations to be the same. Even for those $h$ for which the Hilbert functions or graded Betti numbers of the resulting double point schemes are not uniquely determined, we give one (depending only on $h$) that does occur. We prove the existence of $h^{\max}$, in general, and discuss $h^{\min}$. Our methods include liaison techniques.

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arXiv2003

Higher secant varieties of Segre-Veronese varieties

We study the dimension of the higher secant varieties $X^s$ of ${\Bbb X} = {\Bbb P}^{n_1}\times ...\times {\Bbb P}^{n_t}$ embedded the morphism given by ${\cal O}_{\Bbb X}({a_1,...,a_t})$. We call it a {\it Segre-Veronese variety} and the embedding a {\it Segre-Veronese embedding}. In several cases (e.g. for 3 factors {\Bbb P}^{1}, or when $s$ is small) we show that $X^s$ has the expected dimension except for a few cases. A list of examples of defective $X^s$'s is given.

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arXiv2003

On parameter spaces for artin level algebras

The set of all artin level quotients of a polynomial algebra in n variables having specified socle degree and type admits a parameter space. It is in fact a quasiprojective variety, naturally embedded in a Grassmannian. We give a geometric description of this variety in the case of two variables. Then we give some sufficient conditions for the parameter variety to be projectively normal in the Plucker embedding.

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arXiv2002

Secant Varieties of Grasmann Varieties

In this paper we discuss the dimensions of the (higher) secant varieties to the Grassmann varieties, embedded via the Plucker embeddings. We use Terracini's Lemma and the duality in the exterior algebra of a finite dimensional vector space to translate the problem into that of finding the dimension of a graded piece of a " fat" ideal in the exterior algebra. Among other things, our methods prove that (apart from the chordal varieties to the Grassmannians of lines) all the chordal varieties of Grassmannians have the expected dimension. We also discuss some Grassmannians with deficient secant varieties, giving new proofs that G(3,7) and G(3,9) are of this type. We also find a new Grassmannian, namely G(4,8) with deficient secant varieties.

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