Time: 4:00-5:30 p.m Place: Jeffery Hall 319
Speaker: Gregory G. Smith (Queen's University)
Title: Kemeny’s Proof (Part 2).
Abstract: We continue our examination of Michael Kemeny's approach for describing the minimal free resolution of a canonical curve of even genus. In this second part, we establish the needed vanishing for the appropriate cohomology groups.
Time: 4:30-5:30 p.m Place: Jeffery Hall 319
Speaker: Gregory G. Smith (Queen's University)
Title: Kemeny’s Proof (Part 1).
Abstract: We examine Michael Kemeny's approach for describing the minimal free resolution of a canonical curve of even genus. In the first part, we recast the problem using vector bundles on some auxiliary space.
Time: 4:30-5:30 p.m Place: Jeffery Hall 319
Speaker: Mike Roth (Queen's University)
Title: Lazarsfeld-Mukai bundles.
Abstract: The talk will cover the construction of a particular kind of vector bundle on a K3 surface, obtained by starting with a $g^{1}_{d}$ on a curve on that surface.
This vector bundle construction appeared in Lazarsfeld’s proof of the Petri conjecture, but has also been one of the main starting points for investigating Green’s conjecture. In particular, it will be used in future lectures giving a recent proof due to Kemeny of Green’s conjecture for generic curves of even genus.
Time: 4:30-5:30 p.m Place: Jeffery Hall 319
Speaker: Mike Roth (Queen's University)
Title: K3 surfaces and Lazarsfeld-Mukai bundles.
Abstract: The talk will be a short introduction to some aspects of K3 surfaces, including their relation with canonical embeddings of curves, as well as details of the construction of a particular kind of vector bundle on a K3 surface, obtained by starting with a $g^{r}_{d}$ on a curve on that surface.
This vector bundle construction appeared in Lazarsfeld’s proof of the Petri conjecture, but has also been one of the main starting points for investigating Green’s conjecture. In particular, it will be used in future lectures giving a recent proof due to Kemeny of Green’s conjecture for generic curves of even genus.
Time: 4:00-5:30 p.m Place: Jeffery Hall 319
Speaker: Mike Roth (Queen's University)
Title: An overview of Brill-Noether theory.
Abstract: The object of Brill-Noether theory is to answer the question : which $g^{r}_{d}$’s can a generic curve of genus $g$ have?
The relevance to Green’s conjecture is that this then allows us to compute the Clifford index of a generic curve of genus $g$.
The lecture will be an overview of Brill-Noether theory, including a sketch of the main lines of argument.
Time: 4:00-5:30 p.m Place: Jeffery Hall 319
Speaker: Gregory G. Smith (Queen's University)
Title: Green-Lazarsfeld nonvanishing.
Abstract: By exploiting vector bundle techniques for Koszul cohomology, we see how non-trivial geometry leads to non-trivial syzygies. In particular, this establishes one part of Green’s conjecture.
Time: 4:00-5:30 p.m Place: Jeffery Hall 319
Speaker: Gregory G. Smith (Queen's University)
Title: Koszul cohomology.
Abstract: We introduce and discuss Koszul cohomology from both an algebraic and geometric perspective. This machinery will eventually help us to understand the syzygies of canonical curves.
Time: 1:30-3:00 p.m Place: Jeffery Hall 319
Speaker: Mike Roth (Queen's University)
Title: Special linear series, the Clifford index, and Green’s conjecture.
Abstract: We will introduce the ideas of linear series, and special linear series on a curve, the notation $g^{r}_{d}$, Clifford’s theorem on special linear series, and state Green’s conjecture for canonically embedded curves.
Time: 4:00-5:30 p.m Place: Jeffery Hall 319
Speaker: Gregory G. Smith (Queen's University)
Title: Patterns in the Betti tables.
Abstract: We will examine some of the numerical restrictions on the Betti numbers appearing in the minimal free resolution of the homogeneous coordinate ring of a canonical curve. We will also highlight the consequences of these conditions on low genus curves.
Time: 4:00-5:30 p.m Place: Jeffery Hall 319
Speaker: Mike Roth (Queen's University)
Title: Betti tables for canonical curves of genus 5 and 6, IV.
Abstract: As we do every week, this week we will finish the computation of the possible Betti tables of genus 5 curves.
