Variety of Special Nets of Degree g-1 on Double Coverings of a Smooth Plane Quartic of Genus 9

Abstract

Let C be a complete non-singular curve over ℂ of genus g. We denote by W _d ^r(C) the subscheme of the Picard variety Pic^d(C) whose support is the locus of complete linear series of degree d and dimension at least r. In case d > g + r - 1, W _d ^r (C) = Pic^d(C) and if d = g + r - 1, W _d ^r (C) has dimension d. Therefore the dimension of W _d ^r (C) is independent of C in the range d ≥ g + r - 1. If d ≤ g + r - 2, one knows that dimW _d ^r (C) ≥ p(d, g, r):= g - (r + 1) (g - d + r) for any curve C and is equal to p(d, g, r) for general curve C (see Kleiman-Laksov [9] and Griffiths-Harris [6]). But the dimension of W _d ^r(C) might be greater than p(d, g, r) for some special curve C. Moreover, for curves C with dim W _d ^r (C) > p(d, g, r), C must be of some special type of curves. The first important result along this line is the following well-known theorem of H. Martens which has been extended by a theorem of D. Mumford.

Supported in part by GARC-KOSEF and BSRI #1998–015-D00023.