Gap Sequences and Moduli in Genus 4

Let X denote a compact Riemann surface of genus g > 1. At each point P~X, there is a sequence of g integers, l=71(P)<72(P)<. . .g. Every compact Riemann surface of genus > 1 has a finite number of Weierstrass points. (See [6] for details.) The question of describing the moduli of compact Riemann surfaces of genus g which have a Weierstrass point with a specified first nongap has been answered by Rauch [14] and Arbarello [1]. The problem of describing the moduli of curves of genus g which have a Weierstrass point with a specified gap sequence has been considered by Hensel-Landsberg [7] and Rauch [15] and recently results on this question have been obtained by Pinkham [13] and RimVitulli [16]. In [9], we defined complex spaces of Weierstrass points of the universal curve and obtained a theorem concerning the smoothness and dimension of these spaces. From this, one can recover the moduli results of Rauch [14]. In order to study the higher Weierstrass nongaps and the entire gap sequence, we must extend our definition in [9]. We do this in w and then prove a theorem concerning the smoothness and dimension of these spaces which is analogous to the main result in [9]. F rom this, we can recover a moduli result of Pinkham [13]. In w 3, we study the geometry of these complex spaces of Weierstrass points of the universal curve of genus 4. We obtain a description of the moduli of Teichmiiller surfaces of genus 4 which have a Weierstrass point with a specified gap sequence. In the following, a complex space is not necessarily reduced. If Y is a complex space, then IYI will denote the underlying set of points. If Y, Z1 . . . . . Z , are closed complex subspaces of a complex space, then by Y Z 1 Z 2 . . . Z ,