ON COHOMOLOGY OF THE SQUARE OF AN IDEAL SHEAF

For a smooth subvariety X subset of P-N, consider (analogously to proj ective normality) the vanishing condition H-1(P-N,I-X(2)(k)) = 0, k gr eater than or equal to 3. This condition is shown to be satisfied for all sufficiently large embeddings of a given X, and for a Veronese emb edding of P-n. For C subset of Pg-1, tile canonical embedding of a non -hyperelliptic curve, this condition guarantees the vanishing of some obstruction groups to deformations of the cone. Recall that the tangen ts to deformations are dual to the cokernel of the Gaussian-Wahl map. Theorem. Suppose the Gaussian-Wahl map of C is not surjective and the vanishing condition is fulfilled. Then C is extendable: it is a hyperp lane section of a surface in P-g not the cone over C. Such a surface i s a K-3 if smooth, but it could have serious singularities. Theorem. F or a general curve of genus greater than or equal to 3, this vanishing holds. Conjecture. If the Clifford index is greater than or equal to 3, this vanishing holds.