On the geometry of elliptic pairs
On the geometry of elliptic pairs
复制标题
关于椭圆对的几何
DOI：
10.1016/j.jpaa.2023.107323
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发表时间：
2023
影响因子：
0.8
通讯作者：
Pratt, Elizabeth
中科院分区：
文献类型：
--
作者：
Pratt, Elizabeth
椭圆对 (X, C) 是具有对数终端奇点的射影有理曲面 X，以及包含在 X 的平滑轨迹中的不可约曲线 C，其算术亏格为 1，自交为 0。它们是确定 X 的伪有效锥体是否为多面体的有用工具 [1]，并且本身就是有趣的代数和几何对象。特别令人感兴趣的是环面椭圆对，其中 X 是环面单位元处投影环面的放大。在本文中，我们对所有皮卡德二号环面椭圆对进行分类。令人惊讶的是，事实证明只有其中三个。此外，我们研究了一类非复曲面椭圆对，其来自 P 2 在节点立方体上九个点的放大，其特征为 p。这种构造为我们提供了表面的示例，其中对于一组正密度的素数 p，伪有效锥体是非多面体的，并且假设广义黎曼假设，对于一组正密度的素数 p，伪有效锥体是多面体的。
An elliptic pair (X, C) is a projective rational surface X with log terminal singularities, and an irreducible curve C contained in the smooth locus of X, with arithmetic genus 1 and self-intersection 0. They are a useful tool for determining whether the pseudo-effective cone of X is polyhedral [1], and interesting algebraic and geometric objects in their own right. Especially of interest are toric elliptic pairs, where X is the blow-up of a projective toric surface at the identity element of the torus. In this paper, we classify all toric elliptic pairs of Picard number two. Strikingly, it turns out that there are only three of these. Furthermore, we study a class of non-toric elliptic pairs coming from the blow-up of P 2 at nine points on a nodal cubic, in characteristic p. This construction gives us examples of surfaces where the pseudo-effective cone is non-polyhedral for a set of primes p of positive density, and, assuming the generalized Riemann hypothesis, polyhedral for a set of primes p of positive density.