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.
Time: 4:30-5:30 p.m Place: Jeffery Hall 319
Speaker: Mike Roth (Queen's University)
Title: Betti tables for canonical curves of genus 5 and 6, III.
Abstract: We will continue the computation of the possible Betti tables of genus 5 curves.
Time: 4:30-5:30 p.m Place: Jeffery Hall 319
Speaker: Mike Roth (Queen's University)
Title: Betti tables for canonical curves of genus 5 and 6, II.
Abstract: We will finish the computation of the possible Betti tables of genus 5 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.
Abstract: We will work out the Betti tables for canonically embedded curves of genus 5, and as time permits, genus 6. We will also discuss the classical theorems of Petri and Enriques-Babbage, the ‘basepoint-free pencil trick’ (due to Castelnuovo), and the modern idea of the ‘Castelnovo-Mumford regularity of a sheaf.
Time: 4:00-5:30 p.m Place: Jeffery Hall 319
Speaker: Mike Roth (Queen's University)
Title: Classical results on canonical embeddings II.
Abstract: We will discuss the Hilbert polynomial and Hilbert function of canonically embedded curves, classical theorems of Noether, Castelnuovo, Enriques-Babbage, and Petri, and work out the minimal free resolutions of the homogenous coordinate rings of canonically embedded curves in small genus.
Time: 4:00-5:30 p.m Place: Jeffery Hall 319
Speaker: Mike Roth (Queen's University)
Title: Classical results on canonical embeddings.
Abstract: We will discuss the canonical map in the case of hyperelliptic curves, the geometric interpretation of the Riemann-Roch theorem via the canonical map, the Hilbert polynomial and Hilbert function of canonically embedded curves, and start looking at examples of resolutions of canonical embeddings in small genus.
Time: 4:30-5:30 p.m Place: Jeffery Hall 319
Speaker: Mike Roth (Queen's University)
Title: Canonically embedded curves II.
Abstract: We will show that the canonical map is always an embedding for non-hyperelliptic curves, understand the canonical map in the hyperelliptic case, and discuss the ‘geometric interpretation’ of the Riemann-Roch theorem, via the canonical embedding.
Note that the seminar is starting later than its usual time, and will only go for an hour this week.
Time: 4:00-5:30 p.m Place: Jeffery Hall 319
Speaker: Mike Roth (Queen's University)
Title: Canonically embedded curves.
Abstract: Green’s conjecture concerns ‘canonical curves’ — curves embedded in projective space by their canonical bundle. We will review the basic language of line bundles, divisors, and the Riemann-Roch theorem for curves, and show that, for non-hyperelliptic curves, the canonical bundle of a curve of genus $g\geq 2$ is ‘very ample’, i..e, gives an embedding of the curve in projective space.
