On the moduli space of pointed algebraic curves of low genus. III: Positive characteristic. (Q520670)

The paper deals with moduli space \(\mathcal{M}_{g,1}^N\) of pointed algebraic curves of genus \(g\) with a prescribed semigroup \(N\) whose complement w.r.t. \(\mathbb{N}_0:=\{0,1,2, \cdots\}\) is the gap sequence at the marked point, when the underlying base field is of positive characteristic. In zero characteristic case, there is a nice description of such a moduli space in terms of the negative part of miniversal deformation space of the monomial curve of the semigroup corresponding to the given Weierstrass gap sequence, due to the well-known work of \textit{H. C. Pinkham} [Astérisque 20, 1--131 (1974; Zbl 0304.14006)]. However in positive characteristic case, the isomorphism established by a theorem of Pinkham may not hold in general. In this paper the author establishes criteria for validity of the theorem of Pinkham up to genus 4 in positive characteristic case. Specifically it is proved that Pinkham's statement hold for any numerical semigroup \(N\) of genus \(g\leq 4\) if the characteristic of the base field does not divide any exponent of the generating monomials which defines the negative miniversal deformation of the monomial curve \(X\) w.r.t \(N\), unless \(N=\{5,6,7, 8, \cdots\}\) and \(g=4\). In the case \(N=\{5,6,7, 8, \cdots\}\) (\& \(g=4\)) - which is the semigroup of non-gap sequence at non Weierstrass points - together with additional assumption that the characteristic \(p\) is not five, it is proved that the same statement also holds. This paper has a nice review on the work of Pinkham which is easily accessible to the reader. For part I, II, see [\textit{T. Nakano} and \textit{T. Mori}, ibid. 27, No. 1, 239--253 (2004; Zbl 1077.14037); [\textit{T. Nakano}, ibid. 31, No. 1, 147--160 (2008; Zbl 1145.14025)].