LINEAR SYSTEMS ON IRREGULAR VARIETIES

• Published in: Journal of the Institute of Mathematics of Jussieu Vol. 19; no. 6; pp. 2087 - 2125
• Main Authors: Barja, Miguel Ángel, Pardini, Rita, Stoppino, Lidia
• Format: Journal Article Publication
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.11.2020
• Subjects: 14 Algebraic geometry; 14C Cycles and subschemes; 14J Surfaces and higher-dimensional varieties; Classificació AMS; Continuity (mathematics); Linear systems; Manifolds (Mathematics); Matemàtiques i estadística; Multiplication; Varietats (Matemàtica); Àrees temàtiques de la UPC; irregular variety; 14J35; Clifford-Severi inequalities; 14C20; 14J29; variety of maximal Albanese dimension; continuous rank function; 14J40; eventual map; 14J30
• ISSN: 1474-7480; 1475-3030
• DOI: 10.1017/S1474748019000069

Let $X$ be a normal complex projective variety, $T\subseteq X$ a subvariety of dimension $m$ (possibly $T=X$) and $a:X\rightarrow A$ a morphism to an abelian variety such that $\text{Pic}^{0}(A)$ injects into $\text{Pic}^{0}(T)$; let $L$ be a line bundle on $X$ and $\unicode[STIX]{x1D6FC}\in \text{Pic}^{0}(A)$ a general element. We introduce two new ingredients for the study of linear systems on $X$. First of all, we show the existence of a factorization of the map $a$, called the eventual map of $L$ on $T$, which controls the behavior of the linear systems $|L\otimes \unicode[STIX]{x1D6FC}|_{|T}$, asymptotically with respect to the pullbacks to the connected étale covers $X^{(d)}\rightarrow X$ induced by the $d$-th multiplication map of $A$. Second, we define the so-called continuous rank function$x\mapsto h_{a}^{0}(X_{|T},L+xM)$, where $M$ is the pullback of an ample divisor of $A$. This function 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. As an application, we give quick short proofs of a wide range of new Clifford–Severi inequalities, i.e., geographical bounds of the form $$\begin{eqnarray}\displaystyle \text{vol}_{X|T}(L)\geqslant C(m)h_{a}^{0}(X_{|T},L), & & \displaystyle \nonumber\end{eqnarray}$$ where $C(m)={\mathcal{O}}(m!)$ depends on several geometrical properties of $X$, $L$ or $a$.