We study the number of (set-theoretically) defining equations of Segre products of projective spaces times certain projective hypersurfaces, extending results by Singh and Walther. Meanwhile, we prove some results about the cohomological dimension of certain schemes. In particular, we solve a conjecture of Lyubeznik about an inequality involving the cohomological dimension and the etale cohomological dimension of a scheme, in the characteristic-zero-case and under a smoothness assumption. Furthermore, we show that a relationship between depth and cohomological dimension discovered by Peskine and Szpiro in positive characteristic holds true also in characteristic-zero up to dimension three.
In this paper, we investigate the arithmetical rank of a binomial ideal $J$. We provide lower bounds for the binomial arithmetical rank and the $J$-complete arithmetical rank of $J$. Special attention is paid to the case where $J$ is the binomial edge ideal of a graph. We compute the arithmetical rank of such an ideal in various cases.
Let $I_{G} \subset K[x_{1},...,x_{m}]$ be the toric ideal associated to a finite graph $G$. In this paper we study the binomial arithmetical rank and the $G$-homogeneous arithmetical rank of $I_G$ in 2 cases: $G$ is bipartite, $I_G$ is generated by quadratic binomials. In both cases we prove that the binomial arithmetical rank and the $G$-arithmetical rank coincide with the minimal number of generators of $I_G$.
To any lattice $L \subset \mathbb{Z}^{m}$ one can associate the lattice ideal $I_{L} \subset K[x_{1},...,x_{m}]$. This paper concerns the study of the relation between the binomial arithmetical rank and the minimal number of generators of $I_{L}$. We provide lower bounds for the binomial arithmetical rank and the $\mathcal{A}$-homogeneous arithmetical rank of $I_{L}$. Furthermore, in certain cases we show that the binomial arithmetical rank equals the minimal number of generators of $I_{L}$. Finally we consider a class of determinantal lattice ideals and study some algebraic properties of them…
We study the arithmetical ranks and the cohomological dimensions of an infinite class of Cohen-Macaulay varieties of minimal degree. Among these we find, on the one hand, infinitely many set-theoretic complete intersections, on the other hand examples where the arithmetical rank is arbitrarily greater than the codimension.
In this article we study specializations of multigradings and apply them to the problem of the computation of the arithmetical rank of a lattice ideal $I_{L_{\mathcal{G}}} \subset K[x_{1},...,x_{n}]$. The arithmetical rank of $I_{L_{\mathcal{G}}}$ equals the $\mathcal{F}$-homogeneous arithmetical rank of $I_{L_{\mathcal{G}}}$, for an appropriate specialization $\mathcal{F}$ of $\mathcal{G}$. To the lattice ideal $I_{L_{\mathcal{G}}}$ and every specialization $\mathcal{F}$ of $\mathcal{G}$ we associate a simplicial complex. We prove that combinatorial invariants of the simplicial complex provid…
In this paper we consider reduced homogeneous ideals $\Jcal\subset S$ of a polynomial ring $S$, having a 2-linear resolution. 1. We study systems of generators of $\Jcal\subset S$. 2. We compute the arithmetical rank for a large class of projective curves having a 2-linear resolution. 3. We show that the fiber cone $\proj \Fcal(I_{\Lcal})$ of a lattice ideal $I_{\Lcal}$ of codimension two is a set theoretical complete intersection.
Let $I_M$ and $I_N$ be defining ideals of toric varieties such that $I_M$ is a projection of $I_N$, i.e. $I_N \subseteq I_M$. We give necessary and sufficient conditions for the equality $I_M=rad(I_N+(f_1,...,f_s))$, where $f_1,...,f_s$ belong to $I_M$. Also a method for finding toric varieties which are set-theoretic complete intersection is given. Finally we apply our method in the computation of the arithmetical rank of certain toric varieties and provide the defining equations of the above toric varieties.
Let $M$ be a $(2 \times n)$ non-generic matrix of linear forms in a polynomial ring. For large classes of such matrices, we compute the cohomological dimension (cd) and the arithmetical rank (ara) of the ideal $I_2(M)$ generated by the $2$-minors of $M$. Over an algebraically closed field, any $(2 \times n)$-matrix of linear forms can be written in the Kronecker-Weierstrass normal form, as a concatenation of scroll, Jordan and nilpotent blocks. Bădescu and Valla computed $\mathrm{ara}(I_2(M))$ when $M$ is a concatenation of scroll blocks. In this case we compute $\mathrm{cd}(I_2(M))$ and exten…
In the first part of this paper we study scrollers and linearly joined varieties. A particular class of varieties, of important interest in classical Geometry are Cohen--Macaulay varieties of minimal degree. They appear naturally studying the fiber cone of of a codimension two toric ideals. Let $I\subset S$ be an ideal defining a linearly joined arrangement of varieties: - We compute the depth, and the cohomological dimension. is the connectedness dimension. - We characterize sets of generators of $I$, and give an effective algorithm to find equations, as an application we compute arithmetical…
We give an upper bound for the rank of the border rank 3 partially symmetric tensors. In the special case of border rank 3 tensors $T\in V_1\otimes \cdots \otimes V_k$ (Segre case) we can show that all ranks among 3 and $k-1$ arise and if $\dim V_i \geq 3$ for all $i$'s, then also all the ranks between $k$ and $2k-1$ arise.
Let P^n be the projective space of dimension n on an algebraically closed field of characteristic 0 and F be the image of the Segre embedding of P^1xP^1xP^1 inside P^7. In the present paper we deal with the moduli spaces of locally free sheaves E on F of rank 2 with h^i(F,E(t))=0 for i=1,2 and each integer t.
Let $X$ be a smooth projective hypersurface. In this note we show that any rank 3 arithmetically Cohen-Macaulay vector bundle over $X$ splits when dim $X \geq 7$. We also find a splitting criterion for rank 4 arithmetically Cohen-Macaulay vector bundles on $X$ when dim $X \geq 9$.
We prove the conjecture of Marian-Oprea-Pandharipande on the Segre series associated to a rank zero class. Hence the rank zero Segre integrals on the Hilbert schemes of points for all surfaces are determined.
Recently it has been proved that any arithmetically Cohen-Macaulay (ACM) bundle of rank two on a general, smooth hypersurface of degree at least three and dimension at least four is a sum of line bundles. When the dimension of the hypersurface is three, a similar result is true provided the degree of the hypersurface is at least six. We extend these results to complete intersection subvarieties by proving that any ACM bundle of rank two on a general, smooth complete intersection subvariety of sufficiently high multi-degree and dimension at least four splits. We also obtain partial results in t…
We extend the concept of Segre's Invariant to vector bundles on a surface $X$. For $X=\mathbb{P}^2$ we determine what numbers can appear as the Segre Invariant of a rank $2$ vector bundle with given Chern's classes. The irreducibility of strata with fixed Segre's invariant is proved and its dimensions are computed. Finally, we present applications to the Brill-Noether's Theory for rank $2$ vector bundles on $\mathbb{P}^2.$
We describe the stratification by tensor rank of the points belonging to the tangent developable of any Segre variety. We give algorithms to compute the rank and a decomposition of a tensor belonging to the secant variety of lines of any Segre variety. We prove Comon's conjecture on the rank of symmetric tensors for those tensors belonging to tangential varieties to Veronese varieties.
We study the resolution of an Ulrich bundle of arbitrary rank on the Segre fourfold $\PP^2\times\PP^2$. We characterize the Ulrich bundles $\Vv$ of arbitrary rank on $\PP^2\times\PP^2$ with $h^1(\Vv\otimesΩ\boxtimesΩ)=0$ or with $h^2(\Vv\otimesΩ(-1)\boxtimesΩ(-1))=0$ or obtained as pullback from $\PP^2$ and we construct more complicated examples.
We generalize the results by Eisenbud and Schreyer about Ulrich bundles over Veronese varieties to Segre-Veronese varieties. We discuss the range where we have natural cohomology and we construct multigraded resolutions and monads for Ulrich bundles of any rank. Moreover we give cohomological characterizations for significant families of bundles.
Let S be a nonsingular projective surface. Each vector bundle V on S of rank s induces a tautological vector bundle over the Hilbert scheme of n points of S. When s=1, the top Segre classes of the tautological bundles are given by a recently proven formula conjectured in 1999 by M. Lehn. We calculate here the Segre classes of tautological bundles for all ranks s over all K-trivial surfaces. Furthermore, in rank s=2, the Segre integrals are determined for all surfaces, thus establishing a full analogue of Lehn's formula. We also give conjectural formulas for certain series of Verlinde Euler cha…
We formulate conjectural blowup formulas for Segre and Verlinde numbers on moduli spaces of sheaves on projective surfaces $S$ with $p_g(S)>0$ and $b_1(S)=0$. As applications we give a give a conjectural formula for the Donaldson invariants of $S$ in arbitrary rank, as well as for the $K$-theoretic Donaldson invariants, and some Donaldson invariants with fundamental matters.
In many cases (e.g. for many Segre or Segre embeddings of multiprojective spaces) we prove that a hypersurface of the $b$-secant variety of $X\subset \mathbb {P}^r$ has $X$-rank $>b$. We prove it proving that the $X$-rank of a general point of the join of $b-2$ copies of $X$ and the tangential variety of $X$ is $>b$.
An Ulrich sheaf on an embedded projective variety is a normalized arithmetically Cohen-Macaulay sheaf with the maximum possible number of independent sections. Ulrich sheaves are important in the theory of Chow forms, Boij-Soderberg theory, generalized Clifford algebras, and for an approach to Lech's conjecture in commutative algebra. In this note, we give a reduction of the construction of Ulrich sheaves on a projective variety X to the construction of an Ulrich sheaf for a finite map of curves, which is in turn equivalent to a higher-rank Brill-Noether problem for any of a certain class of c…
An orthogonal bundle over a curve has an isotropic Segre invariant determined by the maximal degree of a Lagrangian subbundle. This invariant, and the induced stratifications on moduli spaces of orthogonal bundles, were studied for bundles of even rank in [4]. In this paper, we obtain analogous results for bundles of odd rank. We obtain a sharp upper bound on the isotropic Segre invariant. We show the irreducibility of the induced strata on the moduli spaces of orthogonal bundles of odd rank, and compute their dimensions. As a key ingredient of the proofs, we study the correspondence between L…
We study virtual invariants of Quot schemes parametrizing quotients of dimension at most 1 of the trivial sheaf of rank N on nonsingular projective surfaces. We conjecture that the generating series of virtual K-theoretic invariants are given by rational functions. We prove rationality for several geometries including punctual quotients for all smooth projective surfaces and dimension 1 quotients for surfaces X with p_g>0. We also show that the generating series of virtual cobordism classes can be irrational. Given a K-theory class on X of rank r, we associate natural series of virtual Segre a…
We define a power series associated with a homogeneous ideal in a polynomial ring, encoding information on the Segre classes defined by extensions of the ideal in projective spaces of arbitrarily high dimension. We prove that this power series is rational, with poles corresponding to generators of the ideal, and with numerator of bounded degree and with nonnegative coefficients. We also prove that this `Segre zeta function' only depends on the integral closure of the ideal. The results follow from good functoriality properties of the `shadows' of rational equivalence classes of projective bund…
Let S be a smooth projective surface equipped with a line bundle H. Lehn's conjecture is a formula for the top Segre class of the tautological bundle associated to H on the Hilbert scheme of points of S. Voisin has recently reduced Lehn's conjecture to the vanishing of certain coefficients of special power series. The first result of this short note is a proof of the vanishings required by Voisin by residue calculations (A. Szenes and M. Vergne have independently found the same proof). Our second result is an elementary solution of the parallel question for the top Segre class on the symmetric…
Polarized varieties without arithmetically Cohen-Macaulay bundles and section rings without graded maximal Cohen-Macaulay modules in characteristic zero
Institute of Mathematics of the Romanian Academy（罗马尼亚科学院数学研究所）
The choice of a homogeneous ideal in a polynomial ring defines a closed subscheme $Z$ in a projective space as well as an infinite sequence of cones over $Z$ in progressively higher dimension projective spaces. Recent work of Aluffi introduces the Segre zeta function, a rational power series with integer coefficients which captures the relationship between the Segre class of $Z$ and those of its cones. The goal of this note is to define a relative version of this construction for closed subschemes of projective bundles over a smooth variety. If $Z$ is a closed subscheme of such a projective bu…
We study entry loci of varieties and their irreducibility from the perspective of $X$-ranks with respect to a projective variety $X$. These loci are the closures of the points that appear in an $X$-rank decomposition of a general point in the ambient space. We look at entry loci of low degree normal surfaces in $\mathbb P^4$ using Segre points of curves; the smooth case was classically studied by Franchetta. We introduce a class of varieties whose generic rank coincides with the one of its general entry locus, and show that any smooth and irreducible projective variety admits an embedding with…
We show that the log canonical bundle, $κ$, of $\bar M_{0,n}$ is very ample, show the homogeneous coordinate ring is Koszul, and give a nice set of rank 4 quadratic generators for the homogeneous ideal: The embedding is equivariant for the symmetric group, and the image lies on many Segre embedded copies of $P^1 \times P^2 \times ... \times P^{n-3}$, permuted by the symmetric group. The homogeneous ideal of $\bar M_{0,n}$ is the sum of the homogeneous ideals of these Segre embeddings.
Determinantal Kernel Schemes of Matrix Nets and Applications to Positive Maps
Troy University（特洛伊大学）；Institute of Mathematics, Vietnam Academy of Science and Technology (VAST)（越南科学技术院数学研究所）；Quy Nhon University（归仁大学）；High School for the Gifted, VNUHCM（越南国立大学胡志明市附属天赋高中）；Vietnam National University Ho Chi Minh City（越南国立大学胡志明市）
We propose a conjecture on the generating series of Chern numbers of tautological bundles on Hilbert schemes of points on curves and establish the rank 1 and rank -1 case of this conjecture. Thus we compute explicitly the generating series of integrals of the Segre classes of tautological bundles of line bundles on curves, which has a similar structure as Lehn's conjecture for surfaces.
We introduce an elementary method to study the border rank of polynomials and tensors, analogous to the apolarity lemma. This can be used to describe the border rank of all cases uniformly, including those very special ones that resisted a systematic approach. We also define a border rank version of the variety of sums of powers and analyse its usefulness in studying tensors and polynomials with large symmetries. In particular, it can be applied to provide lower bounds for the border rank of some very interesting tensors, such as the matrix multiplication tensor. We work in a general setting, …
In this paper, we give necessary and sufficient conditions for the existence of Ulrich bundles on cubic fourfold $X$ of given rank $r$. As consequences, we show that for every integer $r\ge 2$ there exists a family of indecomposable rank $r$ Ulrich bundles on the certain cubic fourfolds, depending roughly on $r$ parameters, and in particular they are of wild representation type; special surfaces on the cubic fourfolds are explicitly constructed by Macaulay2; a new $19$-dimensional family of projective ten-dimensional irreducible holomorphic symplectic manifolds associated to a certain cubic fo…
Given a closed subvariety X in a projective space, the rank with respect to X of a point p in this projective space is the least integer r such that p lies in the linear span of some r points of X. Let W_k be the closure of the set of points of rank with respect to X equal to k. For small values of k such loci are called secant varieties. This article studies the loci W_k for values of k larger than the generic rank. We show they are nested, we bound their dimensions, and we estimate the maximal possible rank with respect to X in special cases, including when X is a homogeneous space or a curv…
For a vector bundle V over a curve X of rank n and for each integer r in the range 1 \le r \le n-1, the Segre invariant s_r is defined by generalizing the minimal self-intersection number of the sections on a ruled surface. In this paper we generalize Lange and Narasimhan's results on rank 2 bundles which related the invariant s_1 to the secant varieties of the curve inside certain extension spaces. For any n and r, we find a way to get information on the invariant s_r from the secant varieties of certain subvariety of a scroll over X. Using this geometric picture, we obtain a new proof of the…
In this paper we contribute to the construction of families of arithmetically Cohen-Macaulay (aCM) indecomposable vector bundles on a wide range of polarized surfaces $(X,\Oo_X(1))$ for $\Oo_X(1)$ an ample line bundle. In many cases, we show that for every positive integer $r$ there exists a family of indecomposable aCM vector bundles of rank $r$, depending roughly on $r$ parameters, and in particular they are of \emph{wild representation type}. We also introduce a general setting to study the complexity of a polarized variety $(X,\Oo_X(1))$ with respect to its category of aCM vector bundles. …
Let $\mathbb X\subset\mathbb P(V)$ be a projective variety, which is not contained in a hyperplane. Then every vector $v$ in $V$ can be written as a sum of vectors from the affine cone $X$ over $\mathbb X$. The minimal number of summands in such a sum is called the rank of $v$. The set of vectors of rank $r$ is denoted by $X_r$ and its projective image by $\mathbb X_r$. The r-th secant variety of $X$ is defined $σ_r(\mathbb X):=\bar{\sqcup_{s\le r}\mathbb X_s}$; it is called tame if $σ_r(\mathbb X)=\sqcup_{s\le r} \mathbb X_s$ and wild if the closure contains elements of higher rank. In this p…
In this PhD thesis we will discuss some aspects in Commutative Algebra which have interactions with Algebraic Geometry, Representation Theory and Combinatorics. In particular, in the first chapter we will focus on understanding when certain cohomology modules vanish, a classical problem raised by Grothendieck. In the second chapter we will use local cohomology to study the connectedness behavior during a Groebner deformation and the arithmetical rank of certain varieties. In the third chapter, we will investigate the relations between the minors of a fixed size of a generic matrix by using too…
