Calabi–Yau threefolds in $\mathbb{P}^n$ and Gorenstein rings
Calabi–Yau threefolds in $\mathbb{P}^n$ and Gorenstein rings
复制标题
$mathbb{P}^n$ 和 Gorenstein 环中的卡拉比-丘三重
DOI：
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发表时间：
2020
影响因子：
1.5
通讯作者：
Beihui Yuan
中科院分区：
文献类型：
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作者：
H. Schenck;M. Stillman;Beihui Yuan
投影正态 Calabi-Yau 三重 $X \subseteq \mathbb{P}^n$ 具有理想的 $I_X$，它在算术上是 Gorenstein，具有 Castelnuovo-Mumford 正则四。当 $I_X$ 是完全交集以及 $X$ 是余维三的情况时，这种理想已经得到了深入研究。在后一种情况下，Buchsbaum-Eisenbud 定理表明 $I_X$ 由斜对称矩阵的 Pfaffians 给出。最近的许多论文研究了当 $I_X$ 具有余维四时的情况。我们证明，对于具有 $\mathrm{codim}(I)=4=\mathrm{reg}(I)$ 的算术 Gorenstein 理想 $I$ 有 16 个可能的贝蒂表，并且其中恰好有 8 个出现在平滑不可约非简并三重中。我们研究余维五或更多的情况，获得 $X$ 与 $h^{p,q}(X)$ 的示例，而不是出现在较低余维 $I_X$ 中的示例或作为复曲面 Fano 簇中的完全交集。我们方法中的一个关键工具是使用逆系统来识别 $X$ 的可能贝蒂表。
A projectively normal Calabi-Yau threefold $X \subseteq \mathbb{P}^n$ has an ideal $I_X$ which is arithmetically Gorenstein, of Castelnuovo-Mumford regularity four. Such ideals have been intensively studied when $I_X$ is a complete intersection, as well as in the case where $X$ is codimension three. In the latter case, the Buchsbaum-Eisenbud theorem shows that $I_X$ is given by the Pfaffians of a skew-symmetric matrix. A number of recent papers study the situation when $I_X$ has codimension four. We prove there are 16 possible betti tables for an arithmetically Gorenstein ideal $I$ with $\mathrm{codim}(I)=4=\mathrm{reg}(I)$, and that exactly 8 of these occur for smooth irreducible nondegenerate threefolds. We investigate the situation in codimension five or more, obtaining examples of $X$ with $h^{p,q}(X)$ not among those appearing for $I_X$ of lower codimension or as complete intersections in toric Fano varieties. A key tool in our approach is the use of inverse systems to identify possible betti tables for $X$.