Linear systems on irregular varieties

Abstract

Let $X$ be a normal complex projective variety, $T\subseteq X$ a subvariety, $a:X\to A$ a morphism to an abelian variety such that ${\mathrm{Pic}}^0(\mathrm{A})$ injects into ${\mathrm{Pic}}^0(\mathrm{T})$ and let $L$ be a line bundle on $X$. Denote by $X^{(d)}\to X$ the connected étale 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 $\alpha \in {\mathrm{Pic}}^0(\mathrm{A})$ general, we study the restricted linear system $|L^{(d)}\otimes a^{\ast }\alpha {|}_{|T^{(d)}}$: if for some $d$ this gives a generically finite map ${\phi }^{(d)}$, we show that f ${\phi }^{(d)}$ is independent of $\alpha$ or $d$ sufficiently large and divisible, and is induced by the {\em eventual map} $\phi :T\to Z$ such that $a_{|T}$ factorizes through $\phi$. The generic value $h_a^0(X_{|T},L)$ 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 $x\mapsto h_a^0(X_{|T},L+xM)$ 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 $${\mathrm{vol}}_{X|T}(\mathrm{L})\ge \mathrm{C}(\mathrm{m}){\mathrm{h}}_{\mathrm{a}}^0({\mathrm{X}}_{|T},\mathrm{L}),$$ where $C(m)=\mathcal{O}(m!)$.

Revised version, 37 pages. The final section has been removed

References in corpus (5)

• Regularity on abelian varieties III: relationship with Generic Vanishing and applications
• The eventual paracanonical map of a variety of maximal Albanese dimension
• Some results on the eventual paracanonical maps
• On the Severi type inequalities for irregular surfaces
• Volume and Hilbert function of R-divisors

Cited by in corpus (8)

• The eventual paracanonical map of a variety of maximal Albanese dimension
• On quint-canonical birationality of irregular threefolds
• Some results on the eventual paracanonical maps
• Relative Clifford inequality for varieties fibered by curves
• Cohomological rank functions on abelian varieties
• Higher Dimensional Slope Inequalities for Irregular fibrations
• Chern degree functions
• Fibered varieties over curves with low slope and sharp bounds in dimension three