Anurag K. Singh, Uli Walther
We compute the arithmetic ranks of the defining ideals of homogeneous coordinate rings of certain Segre products arising from elliptic curves. The cohomological dimension of these ideals varies with the characteristic of the field, though the arithmetic rank does not. We also study the related set-theoretic Cohen-Macaulay property for these ideals.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
When present, the Cohen-Macaulay property can be useful for finding the minimal defining equations of an algebraic variety. It is conjectured that all secant varieties of Segre products of projective spaces are arithmetically Cohen-Macaulay. A summary of the known cases where the conjecture is true is given. An inductive procedure based on the work of Landsberg and Weyman (LW-lifting) is described and used to obtain resolutions of orbits of secant varieties from those of smaller secant varieties. A new computation of the minimal free resolution of the variety of border rank 4 tensors of format…
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 a…
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.
We give an upper bound for the cactus rank of any multi-homogeneous polynomial.
We present some examples of squarefree monomial ideals whose arithmetical rank can be computed using linear algebraic considerations.
We determine a new technique which allows the computation of the arithmetical rank of certain monomial ideals.
Let $I$ be a squarefree monomial ideal of a polynomial ring $S$. In this paper, we prove that the arithmetical rank of $I$ is equal to the projective dimension of $S/I$ when one of the following conditions is satisfied: (1) $μ(I) \leq 5$; (2) $\arithdeg I \leq 4$.
We give an affirmative answer to a question due to J. He and A. Van Tuyl, proving that the arithmetical rank of a special monomial ideal equals to the projective dimension of corresponding quotient module.
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 $I\subset S=K[x_1,...,x_n]$ be a lexsegment edge ideal or the Alexander dual of such an ideal. In both cases it turns out that the arithmetical rank of $I$ is equal to the projective dimension of $S/I.$
We prove some results on the non-existence of rank one maximal Cohen-Macaulay modules over certain Segre product rings. As an application we show that over these Segre product rings there do not exist maximal Cohen-Macaulay modules with multiplicity less than or equal to the parameter degree of the ring, which disproves a conjecture of Schoutens.
We show that for the edge ideals of the graphs consisting of one cycle or two cycles of any length connected through a vertex or a path, the arithmetical rank equals the projective dimension.
When a cone is added to a simplicial complex $Δ$ over one of its faces, we investigate the relation between the arithmetical ranks of the Stanley-Reisner ideals of the original simplicial complex and the new simplicial complex $Δ'$. In particular, we show that the arithmetical rank of the Stanley-Reisner ideal of $Δ'$ equals the projective dimension of the Stanley-Reisner ring of $Δ'$ if the corresponding equality holds for $Δ$.
In this paper, we prove that a squarefree monomial ideal of height 2 whose quotient ring is Cohen-Macaulay is set-theoretic complete intersection.
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 $I_G$.
We show that for the edge ideals of a certain class of forests, the arithmetical rank equals the projective dimension.
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.
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…
Arithmetical rank and cohomological dimension of generalized binomial edge ideals
本文研究连通简单图对应的广义二项式边理想，给出其上同调维数下界，刻画上同调完全交条件，并在多种情形下证明 $J_2(G)$ 的算术秩等于商环投射维数。
We determine sets of elements which, under certain conditions, generate an intersection of ideals up to radical.
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…
We prove that, for the edge ideal of a graph whose cycles are pairwise vertex-disjoint, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes.
We consider the edge ideals of large classes of graphs with whiskers and for these ideals we prove that the arithmetical rank is equal to the big height. Then we extend these results to other classes of squarefree monomial ideals, generated in any degree, proving that the same equality holds.
Let $R$ be a polynomial ring over a field $K$. To a given squarefree monomial ideal $I \subset R$, one can associate a hypergraph $H(I)$. In this article, we prove that the arithmetical rank of $I$ is equal to the projective dimension of $R/I$ when $H(I)$ is a string or a cycle hypergraph.
In this paper we introduce $p-$Ferrer diagram, note that $1-$ Ferrer diagram are the usual Ferrer diagrams or Ferrer board, and corresponds to planar partitions. To any $p-$Ferrer diagram we associate a $p-$Ferrer ideal. We prove that $p-$Ferrer ideal have Castelnuovo mumford regularity $p+1$. We also study Betti numbers, minimal resolutions of $p-$Ferrer ideals. Every $p-$Ferrer ideal is $p-$joined ideals in a sense defined in a fortcoming paper \cite{m2}, which extends the notion of linearly joined ideals introduced and developped in the papers \cite{bm2}, \cite{bm4},\cite{eghp} and \cite{m1…
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.
We study minimal free resolutions of edge ideals of bipartite graphs. We associate a directed graph to a bipartite graph whose edge ideal is unmixed, and give expressions for the regularity and the depth of the edge ideal in terms of invariants of the directed graph. For some classes of unmixed edge ideals, we show that the arithmetic rank of the ideal equals projective dimension of its quotient.
We produce a criterion for open sets in projective $n$-space over a separably closed field to have étale cohomological dimension bounded by $2n-3$. We use the criterion to exhibit a scheme for which étale cohomological dimension is smaller than what a conjecture of G.~Lyubeznik predicts; the discrepancy is of arithmetic nature. For a monomial ideal, we relate extremal graded Betti numbers and étale cohomological dimension of the complement of the corresponding subspace arrangement. Moreover, we derive upper bounds for its arithmetic rank in terms of invariants distilled from the lcm-lattice.
We prove that, for the edge ideal of a cactus graph, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes. The bound is sharp, but in many cases it can be improved. Moreover, we show that the edge ideal of a Cohen-Macaulay graph that contains exactly one cycle or is chordal or has no cycles of length 4 and 5 is a set-theoretic complete intersection.
The arithmetic rank of the residual intersections of a complete intersection ideal
研究完全交理想残差交的算术秩，确定其在任意特征下的上界，并给出其集合理论生成元的显式描述。
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.
The arithmetic rank of nullcones of classical invariant rings
该研究计算特殊线性、正交、辛群作用下零锥理想的算术秩，在特征≠2的无限域上证明其等于不变量环的克鲁尔维数，正特征下用 étale 上同调获取下界。
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…
The arithmetic rank of determinantal nullcones
本文研究了辛群、一般线性群和正交群作用下空锥理想的算术秩及上同调维数，并通过分析临界局部上同调模证明了相关上同调模的消失定理。
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…
Let $R=k[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $k$ and $I$ be a matroidal ideal of degree $d$. In this paper, we study the unmixedness properties and the arithmetical rank of $I$. Moreover, we show that $ara(I)=n-d+1$. This answer to the conjecture that made by H. J. Chiang-Hsieh \cite[Conjecture]{C}.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。