Alternating Catalan numbers and curves with triple ramification

It is known that the monodromy group of each cover of a general curve of genus g>3 equals either the symmetric or the alternating group. The classical Catalan numbers count the minimal degree covers (with symmetric monodromy) of a general curve of even genus. We solve the analogous problem for the alternating group and we determine the number of alternating covers of minimal degree 2g+1 of a general curve of genus g.
18 pages. Minor improvements, also referencing the related work of Lian. Final version to appear in Annali Scuola Normale di Pisa