Mitsuhiro Miyazaki
Let $R$ be the Ehrhart ring of the stable set polytope of a cycle graph which is not Gorenstein. We describe the non-Gorenstein locus of $\mathrm{Spec} R$. Further, we show that $R$ is almost Gorenstein. Moreover, we show that the conjecture of Hibi and Tsuchiya is true.
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In this paper, we give a criterion of the nearly Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph: the Ehrhart ring of the stable set polytope of an h-perfect graph $G$ with connected components $G^{(1)}, \ldots, G^{(\ell)}$ is nearly Gorenstein if and only if (1) for each $i$, the Ehrhart ring of the stable set polytope of $G^{(i)}$ is Gorenstein and (2) $|ω(G^{(i)})-ω(G^{(j)})|\leq 1$ for any $i$ and $j$, where $ω(G^{(i)})$ is the clique number of $G^{(i)}$. We also show that the Segre product of Cohen-Macaulay graded rings with linear non-zerodivisor …
In this paper, we give a criterion of the Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph: the Ehrhart ring of the stable set polytope of an h-perfect graph $G$ is Gorenstein if and only if (1) sizes of maximal cliques are constant (say $n$) and (2) (a) $n=1$, (b) $n=2$ and there is no odd cycle without chord and length at least 7 or (c) $n\geq 3$ and there is no odd cycle without chord and length at least 5.
Let $P$ be a finite poset, $K$ a field, and $O(P)$ (resp. $C(P)$) the order (resp. chain) polytope of $P$. We study the non-Gorenstein locus of $E_K[O(P)]$ (resp. $E_K[C(P)]$), the Ehrhart ring of $O(P)$ (resp. $C(P)$) over $K$, which are each normal toric rings associated $P$. In particular, we show that the dimension of non-Gorenstein loci of $E_K[O(P)]$ and $E_K[C(P)]$ are the same. Further, we show that $E_K[C(P)]$ is nearly Gorenstein if and only if $P$ is the disjoint union of pure posets $P_1, \ldots, P_s$ with $|\mathrm{rank} P_i-\mathrm{rank} P_j|\leq 1$ for any $i$ and $j$.
In this paper, we study nearly Gorensteinness of Ehrhart rings arising from lattice polytopes. We give necessary conditions and sufficient conditions on lattice polytopes for their Ehrhart rings to be nearly Gorenstein. Using this, we give an efficient method for constructing nearly Gorenstein polytopes. Moreover, we determine the structure of nearly Gorenstein (0, 1)-polytopes and characterise nearly Gorensteinness of edge polytopes and graphic matroids.
h-vectors of Gorenstein polytopes
我们表明，Ehrharth-向量的整数Gorenstein多面体与正规幺模三角满足McMullen的g-定理，特别是，它是单峰的。这一结果推广了Athanasiadis（Stanley）最近关于压缩多面体的一个定理.它是从Gorenstein仿射法线幺半群M上的一个更一般的定理导出的：人们可以用一个“长”正则序列因子K[M]（K是一个域），使得其商仍然是正规仿射幺半群代数。这个技巧把关于P的Ehrhart h-向量的所有问题简化为一个Gorenstein多面体Q的Ehrhart h-向量，该多面体Q恰好有一个内部格点，只要在一个多重cP中的每个格点，在N中的c，可以写成P中n个格点的和。如果P有一个正则幺模三角剖分，则很容易得出P的Ehrhart h-向量与一个单纯多胞形的边界复形的组合h-向量重合，并且g-定理适用。
The trace of the canonical module of a Cohen-Macaulay ring describes its non-Gorenstein locus. We study the trace of the canonical module of a Segre product of algebras, and we apply our results to compute the non-Gorenstein locus of toric rings. We provide several sufficient and necessary conditions for Hibi rings and normal semigroup rings to be Gorenstein on the punctured spectrum.
A long-standing open conjecture in combinatorics asserts that a Gorenstein lattice polytope with the integer decomposition property (IDP) has a unimodal (Ehrhart) $h^\ast$-polynomial. This conjecture can be viewed as a strengthening of a previously disproved conjecture which stated that any Gorenstein lattice polytope has a unimodal $h^\ast$-polynomial. The first counterexamples to unimodality for Gorenstein lattice polytopes were given in even dimensions greater than five by Musta{ţ}{ǎ} and Payne, and this was extended to all dimensions greater than five by Payne. While there exist numerous e…
Richard Stanley introduced the order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ arising from a finite partially ordered set $P$, and showed that the Ehrhart polynomial of $\mathcal{O}(P)$ is equal to that of $\mathcal{C}(P)$. In addition, the unimodular equivalence problem of $\mathcal{O}(P)$ and $\mathcal{C}(P)$ was studied by the first author and Nan Li. In the present paper, three integral convex polytopes $Γ(\mathcal{O}(P), -\mathcal{O}(Q))$, $Γ(\mathcal{O}(P), -\mathcal{C}(Q))$ and $Γ(\mathcal{C}(P), -\mathcal{C}(Q))$, where $P$ and $Q$ are partially ordered sets wi…
It is known that every integral convex polytope is unimodularly equivalent to a face of some Gorenstein Fano polytope. It is then reasonable to ask whether every normal polytope is unimodularly equivalent to a face of some normal Gorenstein Fano polytope. In the present paper, it is shown that, by giving new classes of normal Gorenstein Fano polytopes, each order polytope as well as each chain polytope of dimension $d$ is unimodularly equivalent to a facet of some normal Gorenstein Fano polytopes of dimension $d + 1$. Furthermore, investigation on combinatorial properties, especially, Ehrhart …
New elementary components of the Gorenstein locus of the Hilbert scheme of points
本研究构造了希尔伯特函数为$(1,n,n,1)$的不可光滑化Gorenstein代数新显式例子，得到点集希尔伯特概型Gorenstein轨迹的一类新无限基本分支，解决了Iarrobino猜想的三次情形。
Unimodality shenanigans in Ehrhart theory
本文通过构造矩形棱柱的 Cayley 和，否证了 Stanley、Brenti 及 Ferroni-Higashitani 关于 IDP 多胞体 $h^*$-多项式单峰性与对数凹性的多个猜想，并给出光滑反例。
On the Gorenstein locus of some punctual Hilbert schemes
设$k$是代数闭域，$\Hilb_{d}^{G}(\p{N})$是与Gorenstein子概型对应的Hilbert概型$\Hilb_{d}(\p{N})$的开轨迹.我们证明了$\Hilb_{d}^{G}(\p{N})$对于$d\le9$是不可约的，我们几何地刻画了它对于$d\le 8$的奇异性，并给出了当$d=9$时关于它们的一些结果，这些结果为关于$\Hilb_{d}^{G}(\p{N})$中奇点性质的一个猜想提供了一些证据.
The non-Gorenstein locus of stable set rings of finite simple perfect graphs is studied. We describe combinatorially those perfect graphs whose stable set rings are Gorenstein on the punctured spectrum. In addition, we show that, in general, for Cohen--Macaulay graded algebras, their Cohen--Macaulay type and residue are largely independent.
CW-complexes and minimal Hilbert vector of graded Artinian Gorenstein algebras
本文通过几何和拓扑方法研究了格雷德 Artinian Gorenstein 代数的最小 Hilbert 向量，并证明了全 Perazzo 猜想。
A linear generalization of the nearly Gorenstein property, with applications to Veronese subalgebras
本文提出了一种几乎Gorenstein性质的线性推广，应用于Veronese子代数，证明了其在特定条件下的性质。
We analyse the Gorenstein locus of the Hilbert scheme of $d$ points on $\mathbb{P}^n$ i.e. the open subscheme parameterising zero-dimensional Gorenstein subschemes of $\mathbb{P}^n$ of degree $d$. We give new sufficient criteria for smoothability and smoothness of points of the Gorenstein locus. In particular we prove that this locus is irreducible when $d\leq 13$ and find its components when $d = 14$. The proof is relatively self-contained and it does not rely on a computer algebra system. As a by--product, we give equations of the fourth secant variety to the $d$-th Veronese reembedding of $…
In this paper we study the isomorphism classes of local, Artinian, Gorenstein k-algebras A whose maximal ideal M satisfies dim_k(M^3/M^4)=1 by means of Macaulay's inverse system generalizing a recent result by J. Elias and M.E. Rossi. Then we use such results in order to complete the description of the singular locus of the Gorenstein locus of the punctual Hilbert scheme of degree 11.
We study the weak Lefschetz property of artinian Gorenstein algebras and in particular of artinian complete intersections. In codimension four and higher, it is an open problem whether all complete intersections have the weak Lefschetz property. For a given artinian Gorenstein algebra $A$ we ask what linear forms are Lefschetz elements for this particular algebra, i.e., which linear forms $\ell$ give maximal rank for all the multiplication maps $\times \ell: [A]_i \longrightarrow [A]_{i+1}$. This is a Zariski open set and its complement is the \emph{non-Lefschetz locus}. For monomial complete …
Given a commutative noetherian non-positive DG-ring with bounded cohomology which has a dualizing DG-module, we study its regular, Gorenstein and Cohen-Macaulay loci. We give a sufficient condition for the regular locus to be open, and show that the Gorenstein locus is always open. However, both of these loci are often empty: we show that no matter how nice $\mathrm{H}^0(A)$ is, there are examples where the Gorenstein locus of $A$ is empty. We then show that the Cohen-Macaulay locus of a commutative noetherian DG-ring with bounded cohomology which has a dualizing DG-module always contains a de…
Radical property of the traces of the canonical modules of Cohen-Macaulay rings
本文提出介于Gorenstein与Cohen-Macaulay之间的新环性质，证明其在多种环运算下保持，并表明对Schubert循环及某些图环接近Gorenstein。
Geometrical properties of sections of Buchsbaum-Rim sheaves
本文主要讨论高余维算术Gorenstein簇的构造方法。考虑P^n上可分解丛之间的一般一般满射态射G的核层B_G。若B_G具有奇秩，则B_G的正则截面的零轨迹的顶维部分是Gorenstein。我们展示了如何在计算机代数系统Singular中实现该构造方法，并在P^6中生成了一些Gorenstein曲线和三重的例子。最后，我们研究了一类层B_G，其中G的简并轨迹不具有期望的余维数。
We show that the $F$-jumping numbers of a pair $(X, \mathfrak a)$ in positive characteristic have no limit points whenever the symbolic Rees algebra of $-K_X$ is finitely generated outside an isolated collection of points. We also give a characteristic zero version of this result, as well as a generalization of the Hartshorne-Speiser-Lyubeznik-Gabber stabilization theorem describing the non-$F$-pure locus of a variety.
Stanley introduced two classes of lattice polytopes associated to posets, which are called the order polytope ${\mathcal O}_P$ and the chain polytope ${\mathcal C}_P$ of a poset $P$. It is known that, given a poset $P$, the Ehrhart polynomials of ${\mathcal O}_P$ and ${\mathcal C}_P$ are equal to the order polynomial of $P$ that counts the $P$-partitions. In this paper, we introduce the enriched order polytope of a poset $P$ and show that it is a reflexive polytope whose Ehrhart polynomial is equal to that of the enriched chain polytope of $P$ and the left enriched order polynomial of $P$ that…
Deformations of the connected sum of Gorenstein algebras
研究Gorenstein代数连通和的变形，证明Hilbert方案在n≥12时非约化，并构造极化代数的非约化点，揭示分形结构。
Traces of semi-invariants
本文针对有限群不变量环上的半不变量，给出计算其迹的公式，并由此得到不变量环在穿孔谱上为Gorenstein、近Gorenstein以及半不变量局部自由的判据。
On some Gorenstein loci in $\Hilb_{6}(\p 4)$
设k是一个代数闭域，HaG（d）是H（d）（k上射影（d-2）-空间的0维长为d的子概型的Hilbert概型）中对应于算术Gorenstein子概型的开轨迹.我们证明了HaG（6）的不可约性，并刻画了HaG（6）的奇异性.为了实现这些结果，我们也分类所有的阿廷，Gorenstein，不一定分级，k-代数高达6度。此外，我们还描述了HaG（6）中通过某些尺规作图法得到的轨迹.最后证明了HaG（d）中某些点族的障碍性，其中d>5.
The trace of the canonical module (the canonical trace) determines the non-Gorenstein locus of a local Cohen--Macaulay ring. We call a local Cohen--Macaulay ring nearly Gorenstein, if its canonical trace contains the maximal ideal. Similar definitions can be made for positively graded Cohen--Macaulay $K$-algebras. We study the canonical trace for tensor products and Segre products of algebras, as well as of (squarefree) Veronese subalgebras. The results are used to classify the nearly Gorenstein Hibi rings. We study connections between the class of nearly Gorenstein rings and that of almost Go…
Trace ideals of canonical modules over Schubert cycles and determinantal rings
本文研究了施布特环和行列式环的极小迹，给出了非戈林斯坦位的显式描述，并探讨了CTR性质的稳定性与特征。
DG singular equivalence and singular locus
本文从DG奇异范畴构造仿射概形，证明其到奇异轨迹的有限满射，并得出等价DG奇异范畴的环具有相同维数的奇异轨迹。
Symmetry, Unimodality, and Lefschetz Properties for Graded Modules
本文研究极小自由分解为三维广义 Koszul 复形的模的弱 Lefschetz 性质，并推广任意维数有限长度模的非 Lefschetz 轨迹结果。
In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree $1$ for a complete intersection standard Artinian Gorenstein algebra of codimension $6$ presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of "special" Artinian algebras: for them, we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree $1$ and the "lifting" of…
Complete Intersections of Bounded Codimension: Failure of the Nagata Criterion and Recovery of Openness
本文证明余维至多为$c$的完全交性质不满足Nagata准则，构造反例并给出NC的刻画条件，恢复其轨迹开性，还证明相关子集在拟紧优概形上可构造。
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