Twelve points on the projective line, branched covers, and rational elliptic fibrations

Elliptic fibrations on covers of the elliptic modular surface of level 5

We consider the K3 surfaces that arise as double covers of the elliptic modular surface of level 5, $R_{5,5}$. Such surfaces have a natural elliptic fibration induced by the fibration on $R_{5,5}$. Moreover, they admit several other elliptic fibrations. We describe such fibrations in terms of linear systems of curves on $R_{5,5}$. This has a major advantage over other methods of classification of elliptic fibrations, namely, a simple algorithm that has as input equations of linear systems of curves in the projective plane yields a Weierstrass equation for each elliptic fibration. We deal in de…

Xiao的亏格二纤维化：分支覆盖与辫子单值性

Xiao's Genus-Two Fibration: Branched Covers and Braid Monodromy

本文通过分支覆盖构造和辫子单值性方法，直接计算Xiao亏格二Lefschetz纤维化的几何单值性分解，得到类型$(4,3)$的有序正分解及七个因子的Artin坐标。

Van Geemen--Sarti involutions and elliptic fibrations on K3 surfaces double cover of $\mathbb{P}^2$

In this paper we classify the elliptic fibrations on K3 surfaces which are the double cover of a blow up of $\mathbb{P}^2$ branched along rational curves and we give equations for many of these elliptic fibrations. Thus we obtain a classification of the van Geemen--Sarti involutions (which are symplectic involutions induced by a translation by a 2-torsion section on an elliptic fibration) on such a surface. Each van Geemen--Sarti involution induces a 2-isogeny between two K3 surfaces, which is described in this paper.

On hyperelliptic $C^\infty$-Lefschetz fibrations of four-manifolds

We show that hyperelliptic symplectic Lefschetz fibrations are symplectically birational to two-fold covers of rational ruled surfaces, branched in a symplectically embedded surface. This reduces the classification of genus 2 fibrations to the classification of certain symplectic submanifolds in rational ruled surfaces.

一个Mordell-Weil秩为十二的椭圆Calabi-Yau三维流形

An elliptic Calabi-Yau threefold of Mordell-Weil rank twelve

本文构造了一个Mordell-Weil秩为十二的平坦椭圆纤维化Calabi-Yau三维流形，其Hodge数为(25,1)，通过Kummer曲面出发并给出十二个独立截面及crepant解消。

On the geometry of K3 surfaces with finite automorphism group and no elliptic fibrations

Nikulin and Vinberg proved that there are only a finite number of lattices of rank $\geq 3$ that are the Néron-Severi group of projective K3 surfaces with a finite automorphism group. The aim of this paper is to provide a more geometric description of such K3 surfaces $X$, when these surfaces have moreover no elliptic fibrations. In that case we show that such K3 surface is either a quartic with special hyperplane sections or a double cover of the plane branched over a smooth sextic curve which has special tangencies properties with some lines, conics or cuspidal cubic curves. We then study th…