Fano schemes for complete intersections in toric varieties

Abstract: The study of the set of lines contained in a fixed hypersurface is classical: Cayley and Salmon showed in 1849 that a smooth cubic surface contains 27 lines, and Schubert showed in 1879 that a generic quintic threefold contains 2875 lines. More generally, the set of k-dimensional linear spaces contained in a fixed projective variety X itself is called the k-th Fano scheme of X. These Fano schemes have been studied extensively when X is a general hypersurface or complete intersection in projective space.

In this talk, I will report on work with Tyler Kelly in which we study Fano schemes for hypersurfaces and complete intersections in projective toric varieties. In particular, I'll give criteria for the Fano schemes of generic complete intersections in a projective toric variety to be non-empty and of "expected dimension". Combined with some intersection theory, this can be used for enumerative problems, for example, to show that a general degree (3,3)-hypersurface in the Segre embedding of $\mathbb{P}^2\times \mathbb{P}^2$ contains exactly 378 lines.

algebraic geometrynumber theory

Audience: researchers in the topic

SFU NT-AG seminar

Series comments: Description: Research/departmental seminar

Seminar usually meets in-person. For online editions, the Zoom link is shared via mailing list.