Generic Vanishing Index and the Birationality of the Bicanonical Map of Irregular Varieties

On quint-canonical birationality of irregular threefolds

Let $X$ be a complex smooth projective threefold of general type. Assume $q(X)>0$. We show that the $m$-canonical map of $X$ is birational for all $m\geq 5$.

On the bicanonical map of irregular varieties

From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the n…

On surfaces with $p_g=2$, $q=1$ and non-birational bicanonical map

In the present note we show that any surface of general type with $p_g=2$,$q=1$ and non-birational bicanonical map has a pencil of curves of genus 2. Combining this result with previous ones, one obtains thatan irregular surface $S$ of general type with $χ(S)\geq 2$ and non-birational bicanonical map has a pencil of curves of genus 2.

On surfaces with $p_g=q=2$ and non-birational bicanonical map

This paper is devoted to the classification of irregular surfaces of general type with $p_g=q=2$ and non birational bicanonical map. The main result is that, if $S$ is such a surface and if $S$ is minimal with no pencil of curves of genus 2, then $S$ is a double cover of a principally polarized abelian surface $(A,Θ)$, with $Θ$ irreducible. The double cover $S\to A$ is branched along a divisor $B\in |2Θ|$, having at most double points and so $K_S^2=4$.

Cohomological support loci and Pluricanonical systems on irregular varieties

For an irregular variety $X$ of general type, we show that if a general fiber $F$ of the Albanese morphism of $X$ satisfies certain Hodge theoretic condition, the $0$-th cohomological support loci of $K_X$ generates the Picard variety of $X$ . We then show that the condition that the $0$-th cohomological support loci of $K_X$ generates the Picard variety of $X$ can often be applied to prove the birationality of certain pluricanonical maps of $X$.

曲面上点的Hilbert概型的稳定双有理性研究

On the Stable Birationality of Hilbert schemes of points on surfaces

本文研究不规则性为零的曲面上Hilbert概型之间的稳定双有理等价关系，证明几何有理曲面上仅有有限多个稳定双有理类，并由此推出motivic zeta函数的有理性。