Speaker: Nathan Ilten (
https://www.sfu.ca/~nilten/).
Time: 7pm IST, Thursday, 9 July, 2020 (opening time 6:45 pm IST).
Google Meet Link: meet.google.com/qyw-drtx-uvb
Phone: (US) +1 402-744-0304 PIN: 737 931 641#
Title: 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
P2×P2P2×P2 contains exactly 378 lines.