On two monodromy problems for curves in positive characteristic (Q1322064)

\textit{D. Eisenbud} and \textit{J. Harris} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 20, 65-87 (1987; Zbl 0625.14013)] exploited the theory of limit linear series to prove that the monodromy group of the \(g_ d^ 1\)'s on a general curve of genus \(g\) over a field of characteristic zero is the full symmetric group provided the Brill-Noether number \(\rho (g,d, 1)=0\). In the paper under review the following analogous result for curves in positive characteristic is proven: Suppose \(g\geq 3\), \(r>0\) and \(d>0\) are integers such that \(\rho (g,d, r)=0\), and \(K\) and \(L\) are algebraically closed fields with characteristics \(p>0\) and 0 respectively. If the monodromy group of the \(g_ d^ r\)'s on a general curve of genus \(g\) over \(L\) is \(k\)- transitive, then the monodromy group of the \(g_ d^ r\)'s on a general curve of genus \(g\) over \(K\) is \(k\)-transitive as well. Moreover, it is shown that for \(g\geq 2\) the monodromy group of the Weierstrass points on a generic genus \(g\) curve over an algebraically closed field of characteristic \(p> g+1\) is the full symmetric group.