Variety of special linear systems on \(k\)-sheeted coverings
Let \(X\) be a smooth algebraic curve of genus \(g\) which admits a \(k\)-sheeted covering onto a general curve \(C\) of genus \(h > 0\), where \(k\) is an odd prime. In the first section, certain integers \(\varepsilon (h,k)\) and \(\delta (h,k)\) are defined. The authors prove that for \(g \geq \varepsilon(h,k)\), the variety \(W^1_d (X)\) of pencils of degree \(d\) on \(X\) is generically reduced and irreducible with the expected dimension, for all \(d \geq \delta(h,k)\). This result is sharp, in the sense that the authors also show that \(W^1_{\delta (h,k)-1}\) is not irreducible. The authors also discuss the validity of the theorem for \(k=2\) and they discuss sharpness for \(k=3\). In the second section the authors show that for suitable \(d\) and \(g\), and for a general choice of \(X\) as above, there exists an irreducible component of \(W^1_d (X)\) containing a base-point-free \(g^1_d\) which is not composed with the given \(k\)-sheeted covering. Furthermore, every such component is generically smooth of dimension \(\rho(d,g,1)\) (the Brill-Noether number, which is the expected dimension).
scientific article; zbMATH DE number 1127654