Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Curves Seminar

Curves Seminar - Mike Roth (Queen's University)

Wednesday, December 4th, 2019

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.

Curves Seminar - Gregory G. Smith (Queen's University)

Wednesday, November 20th, 2019

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.

Curves Seminar - Mike Roth (Queen's University)

Wednesday, October 23rd, 2019

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.

Curves Seminar - Mike Roth (Queen's University)

Wednesday, October 16th, 2019

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.

Curves Seminar - Mike Roth (Queen's University)

Wednesday, October 9th, 2019

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.

Curves Seminar - Mike Roth (Queen's University)

Wednesday, October 2nd, 2019

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.

Curves Seminar - Mike Roth (Queen's University)

Wednesday, September 25th, 2019

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.

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