Nontrivial rational polynomials in two variables have reducible fibres

On the geometric genus of reducible surfaces and degenerations of surfaces to unions of planes

In this paper we study some properties of degenerations of surfaces whose general fibre is a smooth projective surface and whose central fibre is a reduced, connected surface $X \subset IP^r$, $r \geq 3$, which is assumed to be a union of smooth projective surfaces, in particular of planes. Our original motivation has been a series of papers of G. Zappa which appeared in the 1940-50's regarding degenerations of scrolls to unions of planes. Here, we present a first set of results on the subject; other aspects are still work in progress and will appear later. We first study the geometry and the …

Some stable vector bundles with reducible theta divisors

Let C be a curve of genus g and L a line bundle of degree 2g on C . Let M be the kernel of the evaluation map from the trivial bundle with fibre H^0(C,L) into L . We show that when L is general enough, the rank g bundle M and its exterior powers are stable and admit a reducible theta divisor.

The bicanonical map of surfaces with $p_g=0$ and $K^2\ge 7$, II

We study the minimal complex surfaces of general type with $p_g=0$ and $K^2=7$ or 8 whose bicanonical map is not birational. In the paper 'The bicanonical map of surfaces with $p_g=0$ and $K^2\ge 7$' we have shown that if $S$ is such a surface, then the bicanonical map has degree 2. Here we describe precisely such surfaces showing that there is a fibration $f\colon S\to \pp^1$ such that: i) the general fibre $F$ of $f$ is a genus 3 hyperelliptic curve; ii) the involution induced by the bicanonical map of $S$ restricts to the hyperelliptic involution of $F$. Furthermore, if $K^2_S=8$, then $f$ …

扩展 ADE 曲线的紧化雅可比簇与拉格朗日纤维化

Compactified Jacobians of Extended ADE Curves and Lagrangian Fibrations

本文提出扩展 ADE 曲线概念，描述其紧化雅可比簇分支与相交图的关系，并给出 Beauville-Mukai 型拉格朗日纤维化一般奇异纤维的显式描述。