Abstract. If a curve C is embedded in projective space by a very ample line bundle L , the Gaussian map 4>c L is defined as the pull-back of hyperplane sections of the classical Gauss map composed with the Pliicker 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 GaussianWahl map to be g + 5 for all trigonal curves (i.e., curves which admit a 3-to-l mapping onto the projective line) by examining the way in which a trigonal curve is naturally embedded in a rational normal scroll.

Abstract. In this paper we establish a theorem which determines the invariants of a general hyperplane section of a rational normal scroll of arbitrary dimension. We then construct a complete intersection surface on a fourdimensional scroll and prove it is regular with a trivial dualizing sheaf. We determine the invariants for which the surface is nonsingular, and hence a K3 surface. A general hyperplane section of this surface is a tetragonal curve; we use the first theorem to determine for which tetragonal invariants such a construction is possible. In particular we show that for every genus g ≥ 7 there is a tetragonal curve of genus g that is a hyperplane section of a K3 surface. Conversely, if the tetragonal invariants are not sufficiently balanced, then the complete intersection must be singular. Finally we determine for which additional sets of invariants this construction gives a tetragonal curve as a hyperplane section of a singular canonically trivial surface, and discuss the connection with other recent results on canonically trivial surfaces.

No abstract

Abstract. If a curve C is embedded in projective space by a very ample line bundle L , the Gaussian map 4>c L is defined as the pull-back of hyperplane sections of the classical Gauss map composed with the Pliicker 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 GaussianWahl map to be g + 5 for all trigonal curves (i.e., curves which admit a 3-to-l mapping onto the projective line) by examining the way in which a trigonal curve is naturally embedded in a rational normal scroll.

No abstract