July 6, 2020 - July 10, 2020
Public Access Category: All |
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
- Time:
- 4:00pm-5:30pm
- Description:
- Lecture 1: Monday July 6th, 2020, from 4 pm to 5:30 pm (joining time 3:45pm)
Title: Functorial Geometry and Moduli Spaces
Google Meet Link: https://meet.google.com/wnv-gzos-zga
Abstract: This lecture will introduce some basic concepts of functorial
algebraic geometry following Grothendieck, including flat descent and
representability of functors. A quick introduction will be given to the
moduli problem for vector bundles on curves, and their moduli spaces.
Preparatory reading: An account of the basics of holomorphic vector
bundles on compact Riemann surfaces is available in the 2007 ATM School
lecture notes of Nitsure, which can be downloaded from
https://www.ncmath.org/lecture-notes
- Time:
- 7:00pm
- Description:
- 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.