Linear systems on irregular varieties

TL;DR

This paper investigates the behavior of linear systems on irregular varieties, establishing conditions under which the associated maps are independent of certain parameters and deriving inequalities relating volume and sections.

Contribution

It introduces the concept of the restricted continuous rank and proves its properties, including the extension to a continuous function and explicit derivatives, leading to Clifford-Severi type inequalities.

Findings

The restricted linear systems induce maps independent of parameters for large divisible d.

The restricted continuous rank extends to a continuous, differentiable function with explicit derivatives.

Clifford-Severi inequalities relate volume and sections on smooth irregular varieties.

Abstract

Let $X$ be a normal complex projective variety, $T\subseteq X$ a subvariety, $a\colon X\rightarrow A$ a morphism to an abelian variety such that $\rm{Pic}^0(A)$ injects into $\rm{Pic}^0(T)$ and let $L$ be a line bundle on $X$. Denote by $X^{(d)}\to X$ the connected \'etale cover induced by the $d$-th multiplication map of $A$, by $T^{(d)} \subseteq X^{(d)}$ the preimage of $T$ and by $L^{(d)}$ the pull-back of $L$ to $X^{(d)}$. For $\alpha\in \rm{Pic}^0(A)$ general, we study the restricted linear system $|L^{(d)}\otimes a^*\alpha|_{|T^{(d)}}$: if for some $d$ this gives a generically finite map $\varphi^{(d)}$, we show that f $\varphi^{(d)}$ is independent of $\alpha$ or $d$ sufficiently large and divisible, and is induced by the {\em eventual map} $\varphi\colon T\to Z$ such that $a_{|T}$ factorizes through $\varphi$. The generic value $h^0_a(X_{|T}, L)$ of $h^0(X_{|T}, L\otimes\alpha)$ is called the {\em (restricted) continuous rank.} We prove that if $M$ is the pull back of an ample divisor of $A$, then $x\mapsto h^0_a(X_{|T}, L+xM)$ extends to a continuous function of $x\in\mathbb{R}$, which is differentiable except possibly at countably many points; when $X=T$ we compute the left derivative explicitly. In the case when $X$ and $T$ are smooth, combining the above results we prove Clifford-Severi type inequalities, i.e., geographical bounds of the form $$\rm{vol}_{X|T}(L)\geq C(m) h^0_a(X_{|T},L),$$ where $C(m)={\mathcal O}(m!)$.