1996/01/29 by Jonathan Wahl, Wahl, Jonathan · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.alg-geom/9601027
openalex publication_date 1996/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a smooth subvariety X⊂\Bbb PN, consider (analogously to projective normality) the vanishing condition H1(\Bbb PN,\Cal I2X(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 \Bbb Pn. For C⊂\Bbb Pg-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. \proclaimTheorem 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 Pg not the cone over C.\endproclaim Such a surface is a K3 if smooth, but it could have serious singularities. \proclaimTheorem For a general curve of genus ≥3, this vanishing holds. \endproclaim \proclaimConjecture If the Clifford index is ≥3, this vanishing holds. \endproclaim