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LINEAR SYSTEMS ON IRREGULAR VARIETIES

Part of: Surfaces and higher-dimensional varieties Cycles and subschemes

Published online by Cambridge University Press:  12 March 2019

Miguel Ángel Barja Open the ORCID record for Miguel Ángel Barja [Opens in a new window] ,
Rita Pardini Open the ORCID record for Rita Pardini [Opens in a new window]  and
Lidia Stoppino Open the ORCID record for Lidia Stoppino [Opens in a new window]
Miguel Ángel Barja
Affiliation:
Departament de Matemàtiques, Universitat Politècnica de Catalunya, Avda. Diagonal 647, 08028Barcelona, Spain (miguel.angel.barja@upc.edu)
Rita Pardini
Affiliation:
Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, I-56127 Pisa, Italy (rita.pardini@unipi.it)
Lidia Stoppino
Affiliation:
Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, 27100, Pavia, Italy (lidia.stoppino@unipv.it)
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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

$$\begin{eqnarray}\displaystyle \text{vol}_{X|T}(L)\geqslant C(m)h_{a}^{0}(X_{|T},L), & & \displaystyle \nonumber\end{eqnarray}$$
where $C(m)={\mathcal{O}}(m!)$ depends on several geometrical properties of $X$, $L$ or $a$.

Keywords

irregular varietyvariety of maximal Albanese dimensioneventual mapcontinuous rank functionClifford-Severi inequalities

MSC classification

Primary: 14C20: Divisors, linear systems, invertible sheaves
Secondary: 14J29: Surfaces of general type 14J30: $3$-folds 14J35: $4$-folds 14J40: $n$-folds ($n>4$)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 6 , November 2020 , pp. 2087 - 2125
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Copyright
© Cambridge University Press 2019

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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”.

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