Brill-Noether theory for general branched coverings of \(\mathbb{P}^1\)
Let \({\mathcal M}_g(k)\) be the locus in \({\mathcal M}_g\) of smooth complex curves \(x\) of genus \(g\) such that there exists a point \(P\in X\) with \(h^0(x, {\mathcal O}_X(kP)) \geq 2\) when \(2\leq k\leq g\). It is known that this locus is coreducible and of dimension \(2g-3 +k\) [see for instance \textit{E. Arbarello}, Compos. Math. 29, 325-342 (1974; Zbl 0355.14013)]. The main result of the paper under review is as follows: Suppose \(g\geq 5\), \(k\geq [(g+3)/2]\) and \(g+ 2\leq 2d\leq 2g\). Then for a general element \(X\) in \({\mathcal M}_g(X)\) we have \(\dim W^1_d(X)= \rho(g,1,d) =2d-g-2\) (here \(\rho\) denotes the so-called Brill-Noether number).
scientific article; zbMATH DE number 1734189