Algebraic Geometry Seminar
Jack Huizenga
Penn State
Brill-Noether theory for vector bundles on the projective plane
Abstract: The Brill-Noether theory of curves plays a fundamental role in the
theory of curves and their moduli and has been intensively studied since
the 19th century. In contrast, Brill-Noether theory for vector bundles
and higher dimensional varieties is less understood. It is hard to
determine when Brill-Noether loci are nonempty and these loci can be
reducible and of larger than the expected dimension.
In this talk, we will study Brill-Noether loci for vector bundles on the
projective plane in the case where the number of sections is close to
the largest possible number. When the number of sections is very large,
Brill-Noether problems are all "trivial"--the Brill-Noether loci are
either empty or the entire moduli space. As the number of sections
decreases, we find that there is a "first" nontrivial Brill-Noether
locus, and we discuss its geometry.
Monday September 15, 2025 at 3:00 PM in 636 SEO