The Gaussian-Wahl Map for Trigonal Curves

Abstract

If a curve C is embedded in projective space by a very ample line bundle L, the Gaussian map ${\Phi _{C,L}}$ is defined as the pull-back of hyperplane sections of the classical Gauss map composed with the Plücker embedding. When $L = K$, the canonical divisor of the curve C, the map is known as the Gaussian-Wahl map for C. We determine the corank of the Gaussian-Wahl map to be $g + 5$ for all trigonal curves (i.e., curves which admit a 3-to-1 mapping onto the projective line) by examining the way in which a trigonal curve is naturally embedded in a rational normal scroll.