The combinatorics of degenerate covers and an application for general curves of genus 3. (English) Zbl 1175.14017
Summary: Let \(C_g\) be a general curve of genus \(g\). If \(g\geq4\) then the monodromy group of a primitive cover \(C_g\to\mathbb P^1\) of degree \(n\) is either \(S_n\) or \(A_n\), and both cases actually occur (under suitable conditions on \(n\) for fixed \(g\)). For \(g=3\) also the groups \(\text{GL}_3(2)\) and \({AGL}_3(2)\) occur. In the present paper we settle the last possible case of \({AGL}_4(2)\). This requires new methods (which may be of independent interest) studying the combinatorial structure of degenerate covers.
MSC:
14H30 | Coverings of curves, fundamental group |
14D05 | Structure of families (Picard-Lefschetz, monodromy, etc.) |
14H45 | Special algebraic curves and curves of low genus |