The number of odd ramification maps of degree 2g+1 on a general curve of genus g.

Gian Pietro Pirola
University of Pavia 
Abstract:
Let
\[C\]
be a general complex curve of genus
\[g>2\]
and
\[d=2g+1\]
. It has been proved by Magaard and Völklein that there is a finite number
\[N(g)\]
of maps from
\[C\]
to the projective line having only odd (and then necessarily 3:1) ramification points. These numbers
\[N(g)\]
, that we call the alternate Catalan numbers, are the degrees of dominant maps from Hurwitz spaces of maps to the moduli space of curves. In collaboration with Farkas, Moschetti and Naranjo we give the following:
Theorem. Denote by
\[\sigma_{0,5}\]
and
\[\sigma_{1,4}\]
the Schubert cycles in the Grassmannian of lines in the projective space of dimension
\[2g+1\]
, then
\[N(g)=16^g(\sigma_{0,5}+\sigma_{1,4})^g\]
.
A generating function for the alternate Catalan numbers is also available. The method of the proof involves degeneration (in the Eisenbud-Harris style) to a curve with g-elliptic tails and some de Rham theory.