Linear systems on irregular varieties

Miguel Ángel Barja, Rita Pardini, Lidia Stoppino

Published 2016-06-10Version 1

Let $X$ be a smooth complex projective variety of dimension $n$, let $a\colon X\rightarrow A$ be a map to an abelian variety such that $\dim a(X)=\dim X$ and the induced map $\mathrm{Pic}^0(A)\to \mathrm{Pic}^0(X)$ is injective, and let $L\in \mathrm{Pic}(X)$; denote by $X^{(d)}\to X$ the connected \'etale cover induced by the $d$-th multiplication map of $A$ and by $L^{(d)}$ the pull-back of $L$ to $X^{(d)}$. We study the linear systems $|L^{(d)}\otimes \alpha|$ where $\alpha$ is the pull back of an element of $\mathrm{Pic}^0(A)$. When these systems are non empty for $\alpha$ general, we prove that there exists a map $\varphi\colon X\to Z$ such that $a$ factorizes through $\varphi$ and induces by base change, for $d>>0$ and $\alpha$ general, the map $\varphi^{(d)}$ given by the linear systems $|L^{(d)}\otimes \alpha|$. Let $h^0_a(L)$ be the continuous rank of $L$, i.e., the generic value of $h^0(L\otimes\alpha)$ for $\alpha\in a^*\mathrm{Pic}^0(A)$. We prove that the function $x\mapsto h^0_a(L+xM)$, where $M$ is the pull back of a fixed very ample $H\in \mathrm{Pic}(A)$, extends to a convex differentiable function $\phi(x)$ on $\mathbb R$. We investigate thoroughly the regularity of $\phi$. From these results we also deduce new differentiability properties of the volume function in our setting. Thanks to these results we prove various Clifford-Severi type inequalities of the form $\mathrm{vol}(L)\ge C(n) h^0_a(L)$, extending and refining previous results of the first named author. We also prove Castelnuovo type inequalities, of the form $h^0_a(kL)\ge C(k,n)h^0_a(L)$, where $C(k,n)=\mathcal O(k^n)$. Finally we characterize the triples $(X,a,L)$ in the limit cases of the Clifford-Severi inequalities.