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. |