# Brill–Noether loci for divisors on irregular varieties

Margarida Mendes Lopes; Rita Pardini; Pietro Pirola

Journal of the European Mathematical Society (2014)

- Volume: 016, Issue: 10, page 2033-2057
- ISSN: 1435-9855

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topMendes Lopes, Margarida, Pardini, Rita, and Pirola, Pietro. "Brill–Noether loci for divisors on irregular varieties." Journal of the European Mathematical Society 016.10 (2014): 2033-2057. <http://eudml.org/doc/277525>.

@article{MendesLopes2014,

abstract = {We take up the study of the Brill-Noether loci $W^r(L,X):=\lbrace \eta \in \mathrm \{Pic\}^0(X)\ |\ h^0(L\otimes \eta )\ge r+1\rbrace $, where $X$ is a smooth projective variety of dimension $>1$, $L\in \mathrm \{Pic\}(X)$, and $r\ge 0$ is an integer. By studying the infinitesimal structure of these loci and the Petri map (defined in analogy with the case of curves), we obtain lower bounds for $h^0(K_D)$, where $D$ is a divisor that moves linearly on a smooth projective variety $X$ of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension $>2$. In the $2$-dimensional case we prove an existence theorem: we define a Brill-Noether number $\rho (C, r)$ for a curve $C$ on a smooth surface $X$ of maximal Albanese dimension and we prove, under some mild additional assumptions, that if $\rho (C, r)\ge 0$ then $W^r(C,X)$ is nonempty of dimension $\ge \rho (C,r)$. Inequalities for the numerical invariants of curves that do not move linearly on a surface of maximal Albanese dimension are obtained as an application of the previous results.},

author = {Mendes Lopes, Margarida, Pardini, Rita, Pirola, Pietro},

journal = {Journal of the European Mathematical Society},

keywords = {irregular variety; Brill–Noether theory; Albanese dimension; irregular variety; Brill-Noether theory; Albanese dimension},

language = {eng},

number = {10},

pages = {2033-2057},

publisher = {European Mathematical Society Publishing House},

title = {Brill–Noether loci for divisors on irregular varieties},

url = {http://eudml.org/doc/277525},

volume = {016},

year = {2014},

}

TY - JOUR

AU - Mendes Lopes, Margarida

AU - Pardini, Rita

AU - Pirola, Pietro

TI - Brill–Noether loci for divisors on irregular varieties

JO - Journal of the European Mathematical Society

PY - 2014

PB - European Mathematical Society Publishing House

VL - 016

IS - 10

SP - 2033

EP - 2057

AB - We take up the study of the Brill-Noether loci $W^r(L,X):=\lbrace \eta \in \mathrm {Pic}^0(X)\ |\ h^0(L\otimes \eta )\ge r+1\rbrace $, where $X$ is a smooth projective variety of dimension $>1$, $L\in \mathrm {Pic}(X)$, and $r\ge 0$ is an integer. By studying the infinitesimal structure of these loci and the Petri map (defined in analogy with the case of curves), we obtain lower bounds for $h^0(K_D)$, where $D$ is a divisor that moves linearly on a smooth projective variety $X$ of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension $>2$. In the $2$-dimensional case we prove an existence theorem: we define a Brill-Noether number $\rho (C, r)$ for a curve $C$ on a smooth surface $X$ of maximal Albanese dimension and we prove, under some mild additional assumptions, that if $\rho (C, r)\ge 0$ then $W^r(C,X)$ is nonempty of dimension $\ge \rho (C,r)$. Inequalities for the numerical invariants of curves that do not move linearly on a surface of maximal Albanese dimension are obtained as an application of the previous results.

LA - eng

KW - irregular variety; Brill–Noether theory; Albanese dimension; irregular variety; Brill-Noether theory; Albanese dimension

UR - http://eudml.org/doc/277525

ER -

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