Article contents
LINEAR SYSTEMS ON IRREGULAR VARIETIES
Published online by Cambridge University Press: 12 March 2019
Abstract
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





Keywords
MSC classification
- Type
- Research Article
- Information
- Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 6 , November 2020 , pp. 2087 - 2125
- Copyright
- © Cambridge University Press 2019
Footnotes
The first author was supported by MINECO MTM2015-69135-P “Geometría y Topología de Variedades, Álgebra y Aplicaciones” and by Generalitat de Catalunya SGR2014-634. The second and third authors are members of G.N.S.A.G.A.–I.N.d.A.M. This research was partially supported by MIUR (Italy) through PRIN 2010–11 “Geometria delle varietà algebriche” and PRIN 2012–13 “Moduli, strutture geometriche e loro applicazioni”.
References
- 13
- Cited by