Algebraic and combinatorial Brill-Noether theory
type: Item
instance of: scholarly article
zbMATH Open document ID: 1262.14031
author: Lucia Caporaso
publication date: 2012-07-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1106.1140
Mathematics Subject Classification ID: 14H51
Mathematics Subject Classification ID: 05Cxx
Mathematics Subject Classification ID: 14D06
zbMATH DE Number: 6052324
zbMATH Keywords: algebraic curves
zbMATH Keywords: divisors
zbMATH Keywords: Brill-Noether theory
zbMATH Keywords: graph
MaRDI profile type: Publication
arXiv ID: 1106.1140
arXiv classification: math.AG
arXiv classification: math.CO
title: Algebraic and combinatorial Brill-Noether theory
review text: This paper covers a particular topic in the interplay between the divisor theory on algebraic curves and the corresponding theory on finite graphs: the Brill--Noether Theory.NEWLINENEWLINERecall that, if \(C\) is an algebraic curve of genus \(g\) over an algebraically closed field, then for any couple of positive integers \(r\) and \(d\), one defines the Brill-Noether variety as NEWLINE\[NEWLINE W^r_d(C) := \left\{L \in \mathrm{Pic}^dC : r(C,L) \geq r \right\} NEWLINE\]NEWLINE where \(\mathrm{Pic}^dC\) denotes the set of invertible sheaves of degree \(d\) on \(C\) and \(r(C,L):= {\dim}(H^0(C,L)) -1\); and the Brill--Noether number as NEWLINE\[NEWLINE \rho^r_d(g) := g-(r+1)(g-d+r). NEWLINE\]NEWLINE The link between them has been widely studied and the first remarkable results are the following (see [\textit{G. Kempf}, Schubert methods with an application to algebraic curves. Math. Centrum, Amsterdam, Afd. zuivere Wisk. ZW 6/71 (1971; Zbl 0223.14018)], \textit{S. L. Kleiman} and \textit{D. Laksov} [Am. J. Math. 94, 431--436 (1972; Zbl 0251.14005), Acta Math. 132, 163--176 (1974; Zbl 0286.14005)], \textit{P. Griffiths} and \textit{J. Harris} [Duke Math. J. 47, 233--272 (1980; Zbl 0446.14011)]):NEWLINENEWLINEExistence Theorem. If \(\rho^r_d(g) \geq 0\) and \(C\) is a smooth curve of genus \(g\), then \(W^r_d(C) \neq \emptyset\).NEWLINENEWLINEBrill-Noether Theorem. If \(\rho^r_d(g) < 0\) and \(C\) is a \textit{general} smooth curve of genus \(g\), then \(W^r_d(C) = \emptyset\).NEWLINENEWLINE In this paper a rather synthetic overview of the divisor theory on graphs is given. In analogy with the algebro--geometric setting, the notion of divisor (its genus, its degree, the intersection law, the Jacobian group, a ``combinatorial'' rank) on a finite connected graph \(\Gamma\) is reminded. So one can define \(W^r_d(\Gamma)\) as the set of the (classes of) divisors of degree \(d\) having combinatorial rank not exceeding \(r\). One should say that the notion of combinatorial ran…
reviewed by: Michela Brundu
description: scientific article; zbMATH DE number 6052324