In his classical work, Pinkham discovered a beautiful theorem on the moduli space of pointed algebraic curves with a fixed Weierstrass gap sequence at the marked point. Namely, the complement of a Weierstrass gap sequence in the set of non-negative integers is a numerical semigroup, and he described such a moduli space in terms of the negative part of the miniversal deformation space of the monomial curve of this semigroup. Unfortunately, his theorem holds only in characteristic 0 and does not hold in positive characteristic in general. In this paper, we will study his theorem in positive characteristic, and give a fairly sharp condition for his theorem to hold in positive characteristic up to genus 4. As an application, we present a complete analysis of his theorem in positive characteristic in the low genus case.
"On the Moduli Space of Pointed Algebraic Curves of Low Genus III ---Positive Characteristic---." Tokyo J. Math. 39 (2) 565 - 582, December 2016. https://doi.org/10.3836/tjm/1484903137