LINEAR SYSTEMS ON IRREGULAR VARIETIES
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{P...
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| Published in: | Journal of the Institute of Mathematics of Jussieu Vol. 19; no. 6; pp. 2087 - 2125 |
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| Main Authors: | , , |
| Format: | Journal Article Publication |
| Language: | English |
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Cambridge, UK
Cambridge University Press
01.11.2020
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| ISSN: | 1474-7480, 1475-3030 |
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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 $a$. |
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| AbstractList | Peer Reviewed 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 $a$. 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 \(a\). 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 $a$ . |
| Author | Pardini, Rita Barja, Miguel Ángel Stoppino, Lidia |
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| Cites_doi | 10.1353/ajm.0.0054 10.4310/PAMQ.2008.v4.n3.a1 10.1007/978-3-662-06307-1 10.1007/978-3-642-18810-7 10.4310/jdg/1214438689 10.4007/annals.2013.177.3.6 10.1007/s00222-004-0399-7 10.1007/BF01450841 10.1093/imrn/rnx127 10.1007/978-1-4757-5323-3 10.1353/ajm.2011.0000 10.1090/S1056-3911-08-00490-6 10.1007/s00208-007-0146-7 10.1007/s00208-014-1025-7 10.24033/asens.2109 10.1007/BFb0090889 10.1007/BF01202721 10.1007/s13348-016-0169-z 10.1090/S0894-0347-97-00223-3 10.1215/00127094-2871306 10.1002/mana.200310115 10.1007/BF01388711 10.1016/j.matpur.2015.11.012 10.24033/bsmf.2508 10.1007/978-3-642-18808-4 |
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| Contributor | Universitat Politècnica de Catalunya. GEOMVAP - Geometria de Varietats i Aplicacions Universitat Politècnica de Catalunya. Departament de Matemàtiques |
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| Snippet | 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... 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... 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... Peer Reviewed |
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| SubjectTerms | 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 |
| Title | LINEAR SYSTEMS ON IRREGULAR VARIETIES |
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