Margherita Lelli-Chiesa

ABSTRACT

A projective variety in an $n$-dimensional projective space is extendable if it can be realized as a hyperplane section in a projective space of dimension $n+1$ of a variety that is not a cone. In the 1990s, Wahl reduced the extendability problem for canonical curves with non-surjective Wahl map to a vanishing statement, using in a crucial way the deformation theory of the affine cone. This result was later fundamental in the characterization of hyperplane sections of K3 surfaces achieved about twenty years later by Arbarello, Bruno, and Sernesi. I will present an alternative proof of Wahl’s theorem based on multiple curves, that is, non-reduced structures supported on smooth curves, and discuss applications to the more general question of whether a curve in projective space lies on a surface embedded in the same space.

This is joint work with Arbarello and Bruno.