Calabi-Yau 型簇的整体截面与双有理几何

中文摘要 AI 辅助

我们研究了除子的整体截面与双有理几何之间关系的若干方面。我们证明，对于 klt Calabi-Yau 型簇 $X$，整体截面由控制 $D$-极小模型纲领（$D$-MMP）的同一双有理数据所支配。因此，任意大除子的整体截面可以利用消没定理系统地约化为在合适的双有理模型 $Y$ 上的欧拉示性数 $\chi(Y,\mathcal O_Y(D))$，从而将经典的基于正性的方法推广到移动锥之外。该约化过程通过移除固定的除子分量（这些分量对整体截面空间没有贡献）来实现，既可以直接通过减法，也可以通过双有理收缩到合适的 $D$-极小模型。我们在大切锥上获得了整体截面的分段拟多项式公式，其定义域为 $D$-MMP 的 Mori 室。限制到 Fano 型簇的子类时，这些公式可以推广到整个有效锥。这些公式仅由 $X$ 的小双有理类即可构造，却编码了 $X$ 的所有双有理收缩的性质。除了从双有理几何计算整体截面之外，我们还研究了逆过程，即有限次计算 $h^0(X,\mathcal O_X(D))$ 可以确定室分解及相关双有理模型的若干方面。

英文摘要

We investigate aspects of the relationship between global sections of divisors and birational geometry. We show that for varieties $X$ of klt Calabi-Yau type, global sections are governed by the same birational data that control the $D$-minimal model program ($D$-MMP). As a consequence, global sections of arbitrary big divisors can be systematically reduced to Euler characteristics $χ(Y,\mathcal O_Y(D))$ on suitable birational models $Y$ using vanishing theorems, extending classical positivity-based methods beyond the movable cone. The reduction proceeds by removing fixed divisorial components, which do not contribute to the space of global sections, either directly by subtraction or birationally by contraction to a suitable $D$-minimal model. We obtain piecewise quasipolynomial formulae for global sections on the big cone, whose domains are the Mori chambers of the $D$-MMP. Restricting to the subclass of Fano type varieties extends such formulae to the entire effective cone. These formulae can be constructed from just the small birational class of $X$, yet encode properties of all birational contractions of $X$. Alongside the computation of global sections from birational geometry, we study the inverse process, wherein finite computations of $h^0(X,\mathcal O_X(D))$ determine aspects of the chamber decomposition and the associated birational models.

发表机构

• University of Birmingham（伯明翰大学）
• University of Oxford（牛津大学）
• Cornell University（康奈尔大学）

机构由 AI 辅助整理，请以论文原文为准。

补充信息

备注

43 pages