论文标题
$ \ mathbb {p}^n $和gorenstein戒指的calabi-yau三倍
Calabi-Yau threefolds in $\mathbb{P}^n$ and Gorenstein rings
论文作者
论文摘要
一个普通的普通calabi-yau三倍$ x \ subseteq \ mathbb {p}^n $具有理想的$ i_x $，这是Castelnuovo-Mumford Quaranity Quaranity Quaranity Quaranity for的算术上的Gorenstein。当$ i_x $是一个完整的交叉点时，以及在$ x $是codimension三的情况下，就对这些理想进行了深入的研究。在后一种情况下，buchsbaum-eisenbud定理表明$ i_x $由偏斜矩阵的pfaffians给出。 $ i_x $具有编成四个的$ i_x $时，许多最近的论文研究了这种情况。我们证明，有16个可能的Betti表用于算术上的Gorenstein理想$ i $，$ \ Mathrm {codim}（i）= 4 = 4 = \ Mathrm {reg}（i）$，其中8个正好出现了，对于平滑的不可偿还的不可偿还的不可修复的不符号。我们调查了Condimension 5或更多的情况下的情况，并获得了$ h^{p，Q}（x）$的$ x $的示例，而不是以$ i_x $出现的$ i_x $的较低的consimimension或toric fano品种中的完整交集。我们方法中的一个关键工具是使用反系统来确定$ x $的可能的betti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 $X$.
