On Cohomology of the Square of an Ideal Sheaf

For a smooth subvariety $X\subset\Bbb P^N$, consider (analogously to projective normality) the vanishing condition $H^1(\Bbb P^N,\Cal I^2_X(k))=0$, $k\ge3$. This condition is shown to be satisfied for all sufficiently large embeddings of a given $X$, and for a Veronese embedding of $\Bbb P^n$. For $C\subset\Bbb P^{g-1}$, 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 Gaussian-Wahl map. \proclaim{Theorem} Suppose the Gaussian-Wahl map of $C$ is not surjective and the vanishing condition is fulfilled. Then $C$ is {\bf extendable}: it is a hyperplane section of a surface in $\Bbb P^g$ not the cone over $C$.\endproclaim Such a surface is a K3 if smooth, but it could have serious singularities. \proclaim{Theorem} For a general curve of genus $\ge3$, this vanishing holds. \endproclaim \proclaim{Conjecture} If the Clifford index is $\ge3$, this vanishing holds. \endproclaim