Algebraic and combinatorial Brill-Noether theory
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 rank is more subtle when \(\Gamma\) contains loops: it is necessary to introduce ``refinements'' of \(\Gamma\) as graphs with further vertices on its loops.NEWLINENEWLINESince to a nodal curve \(X\) one can associate the dual graph, whose vertices (respectively, edges) represent the irreducible components of \(X\) (resp. the nodes of \(X\)), the study of the analogous for graphs of the two theorems above is very natural.NEWLINENEWLINENEWLINENEWLINE The main result of this paper is the Existence Theorem for graphs which claims that, if \(\rho^r_d(g) \geq 0\), then for every graph \(\Gamma\) of genus \(g\) it holds \(W^r_d(\Gamma) \neq \emptyset\). The proof skillfully uses the cited corresponding result for algebraic curves and a suitable version of the Baker specialization Lemma. But the problem of finding a purely combinatorial proof of such result still stands.NEWLINENEWLINEFinally, the author observes that a sort of Brill--Noether Theorem for graphs also holds: if \(\rho^r_d(g) < 0\) then there exists a graph \(\Gamma\) of genus \(g\) such that \(W^r_d(\Gamma) = \emptyset\) (this follows from a result of \textit{F. Cools} et al., [Adv. Math. 230, No. 2, 759--776 (2012; Zbl 1325.14080)]).NEWLINENEWLINEHence the author conjectures that some assumption on a specific graph \(\Gamma\) could imply \(W^r_d(\Gamma) = \emptyset\), provided that \(\rho^r_d(g) < 0\).NEWLINENEWLINEThe final interesting result deeply relates the subjects of the two Brill--Noether Theorems: if there exists a graph \(\Gamma\) of genus \(g\) such that \(W^r_d(\Gamma) = \emptyset\) then \(W^r_d(C) = \emptyset\) for a general projective curve \(C\).NEWLINENEWLINEIn particular, if \(\rho^r_d(g) < 0\) then the Brill--Noether Theorem for curves holds.NEWLINENEWLINEFor the entire collection see [Zbl 1236.14001].
scientific article; zbMATH DE number 6052324