On Cohomology of the Square of an Ideal Sheaf

For a smooth subvariety X ⊂ P , consider (analogously to projective normality) the vanishing condition H(P , I X (k)) = 0, k ≥ 3. This condition is shown to be satisfied for all sufficiently large embeddings of a given X, and for a Veronese embedding of P. For C ⊂ P, the canonical embedding of a non-hyperelliptic curve, this condition guarantees the vanishing of some obstruction groups to deformations of the cone. Recall that the tangents to deformations are dual to the cokernel of the GaussianWahl 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 hyperplane section of a surface in P not the cone over C. Such a surface is a K3 if smooth, but it could have serious singularities. Theorem. For a general curve of genus ≥ 3, this vanishing holds. Conjecture. If the Clifford index is ≥ 3, this vanishing holds. 0. Introduction Let L be a very ample line bundle on a smooth complex projective variety X , giving an embedding X ⊂ P . It is well-known that projective normality of the embedding (or normal generation of L) is equivalent to the vanishing H(P , IX(k)) = 0, all k, where IX is the ideal sheaf defining X ; further, all sufficiently high powers of L are normally generated. In this paper, we shall be concerned with the condition on L (or its embedding) (∗) H(P , I X(k)) = 0, all k 6= 2. (For k=2, Proposition 1.8 shows this group is frequently the kernel of the Gaussian map of L, hence is rarely 0.) This question arises naturally because these cohomology groups give the torsion submodule of the Kähler differentials of the affine cone over X (Proposition 1.4). Our first main results are: 1