论文标题
关于由togliatti系统参数的品种的算术cohen-macaulayness
On the arithmetic Cohen-Macaulayness of varieties parameterized by Togliatti systems
论文作者
论文摘要
给定任何对角线环状亚组$λ\子集gl（n+1，k）$ d $的$ d $，令$ i_d \ subset k [x_0，\ ldots，x_n] $是所有单元$ \ {m_ {1}，\ ldots，\ ldots，\ ldots，m_ {rd $ d $ d $ d $ $λ$。 $ i_d $是一个单一的togliatti系统，提供了$ r \ leq \ binom {d+n-1} {n-1} $，在这种情况下，在这种情况下是$（m_ {1}，\ ldots，\ ldots，m_ {r}）$ gt $ gt $ gt $ gt $ gt $ gt $ - 我们证明所有这些$ gt $ varieties都是偶然的Cohen-Macaulay，我们给出了Hilbert功能的组合表达。在情况下，$ n = 2 $，我们明确计算Hilbert功能，多项式和$ x_d $系列。我们确定了其同质理想的最小自由分辨率，我们表明它是四边形和立方体产生的二项式理想。我们还提供两种类型的发电机的确切数量。最后，我们提出了确定由Togliatti系统参数化的表面是否为ACM的问题。我们构建了ACM的示例和非ACM的示例。
Given any diagonal cyclic subgroup $Λ\subset GL(n+1,k)$ of order $d$, let $I_d\subset k[x_0,\ldots, x_n]$ be the ideal generated by all monomials $\{m_{1},\ldots, m_{r}\}$ of degree $d$ which are invariants of $Λ$. $I_d$ is a monomial Togliatti system, provided $r \leq \binom{d+n-1}{n-1}$, and in this case the projective toric variety $X_d$ parameterized by $(m_{1},\ldots, m_{r})$ is called a $GT$-variety with group $Λ$. We prove that all these $GT$-varieties are arithmetically Cohen-Macaulay and we give a combinatorial expression of their Hilbert functions. In the case $n=2$, we compute explicitly the Hilbert function, polynomial and series of $X_d$. We determine a minimal free resolution of its homogeneous ideal and we show that it is a binomial prime ideal generated by quadrics and cubics. We also provide the exact number of both types of generators. Finally, we pose the problem of determining whether a surface parameterized by a Togliatti system is aCM. We construct examples that are aCM and examples that are not.
