Rolling factors deformations and extensions of canonical curves

Abstract

A tetragonal canonical curve is the complete intersection of two divisors on a scroll. The equations can be written in 'rolling factors' format. For such homogeneous ideals we give methods to compute inflnitesimal deformations. Deformations can be obstructed. For the case of quadratic equations on the scroll we derive explicit base equations. They are used to study extensions of tetragonal curves. 2000 Mathematics Subject Classiflcation: 14B07 14H51 14J28 32S30 Keywords and Phrases: Tetragonal curves, rolling factors, K3 surfaces An easy dimension count shows that not all canonical curves are hyperplane sections of K3 surfaces. A surface with a given curve as hyperplane section is called an extension of the curve. With this terminology, the general canonical curve has only trivial extensions, obtained by taking a cone over the curve. In this paper we concentrate on extensions of tetragonal curves. The extension problem is related to deformation theory for cones. This is best seen in terms of equations. Suppose we have coordinates (x0 :¢¢¢ : xn : t) on P n+1 with the special hyperplane section given by t = 0. We describe an extension W of a variety V :fj(xi) = 0 by a system of equations Fj(xi;t) = 0 with Fj(xi; 0) = fj(xi). We write Fj(xi;t) = fj(xi) + tf 0 j(xi) +¢¢¢ + ajt d j ,