论文标题
在椭圆对的几何形状上
On The Geometry Of Elliptic Pairs
论文作者
论文摘要
椭圆对$（x，c）$是带有日志终端奇点的投影理性表面$ x $，而在$ x $的平滑基因座中包含的不可约束曲线$ c $，算术属属于零属和自我解干。它们是确定$ x $的伪有效锥体是多面体的，并且本身是否具有有趣的代数和几何对象。尤其是感兴趣的是圆环椭圆对，其中$ x $是在圆环的身份元素处弹出的紫杉表面的爆炸。在本文中，我们将所有PICARD第二的复曲椭圆形对分类。令人惊讶的是，事实证明其中只有三个。此外，我们研究了一类非椭圆对，来自$ \ mathbb {p}^2 $的爆炸，在鼻子立方的九点，特征性$ p $。这种结构为我们提供了表面的例子，其中伪有效的锥体对于一组正密度的素数$ p $是非多层的，并且假设有一套普遍的Riemann假设，则用于一组Primes $ p $正密度的Primes $ P $。
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 one and self-intersection zero. They are a useful tool for determining whether the pseudo-effective cone of $X$ is polyhedral, 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 $\mathbb{P}^2$ 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.
