On covering of curves and non-Weierstrass semigroups

Fernando Torres
University of Campinas
Abstract:
Let
\[X\]
be a (projective, irreducible, non-singular algebraic) curve defined over an algebraically closed field. Let
\[\mathbb{N}_0\]
be the monoid of non-negative integers with the operation of addition. To any point
\[P\in X\]
there is associated the set 
\[H(P)\]
of pole orders of regular functions on
\[X\setminus \{P\}\]
which is a submonoid of
\[\mathbb{N}_0\]
with
\[\mathbb{N}_0\setminus H(P)\]
finite. Conversely given a submonoid
\[H\subseteq\mathbb{N}_0\]
with finite complement in
\[\mathbb{N}_0\]
, is there a pointed curve
\[(X, P)\]
such that 
\[H=H(P)\]
? This was an outstanding question posed by Hurwitz around 1893 which remained open for a long time. Indeed, Buchweitz (around 1980) solved this question in a negative way. In this talk we examine Hurwitz’s question by means of certain covering of curves.