$\mathbb{P}^n$和Gorenstein环中的Calabi-Yau三倍

IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Advances in Theoretical and Mathematical Physics Pub Date : 2020-11-21 DOI:10.4310/atmp.2022.v26.n3.a7
H. Schenck, M. Stillman, Beihui Yuan
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引用次数: 2

摘要

一个射射正规的Calabi-Yau三倍$X \subseteq \mathbb{P}^n$具有理想$I_X$,它在算术上是Castelnuovo-Mumford正则性4的Gorenstein。当$I_X$是一个完全相交时,以及$X$是余维三的情况下,这些理想已经被深入研究。在后一种情况下,Buchsbaum-Eisenbud定理表明$I_X$是由偏对称矩阵的Pfaffians给出的。最近的一些论文研究了$I_X$具有余维四的情况。我们证明了$\mathrm{codim}(I)=4=\mathrm{reg}(I)$的算术Gorenstein理想$I$有16个可能的betti表,而对于光滑不可约非退化三折,正好有8个可能的betti表。我们研究了余维数大于等于5的情况,得到了$X$具有$h^{p,q}(X)$的例子,这些例子不属于低余维数$I_X$出现的例子,也不属于环型Fano变体中的完全交点。我们的方法中的一个关键工具是使用逆系统来识别$X$可能的betti表。
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Calabi–Yau threefolds in $\mathbb{P}^n$ and Gorenstein rings
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 $X$.
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来源期刊
Advances in Theoretical and Mathematical Physics
Advances in Theoretical and Mathematical Physics 物理-物理:粒子与场物理
CiteScore
2.20
自引率
6.70%
发文量
0
审稿时长
>12 weeks
期刊介绍: Advances in Theoretical and Mathematical Physics is a bimonthly publication of the International Press, publishing papers on all areas in which theoretical physics and mathematics interact with each other.
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