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, 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.

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

Wednesday, September 18th, 2019

Time: 4:00-5:30 p.m.   Place: Jeffery Hall 319

Speaker: Gregory G. Smith (Queen's University)

Title: Koszul complexes

Abstract: We introduce Koszul complexes and examine a couple different interpretations.  Beyond providing a concrete family of minimal free resolutions, this structure plays a significant role in analyzing the syzygies of canonical curves.

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

Wednesday, September 11th, 2019

Time: 4:00-5:30 p.m.   Place: Jeffery Hall 319

Speaker: Gregory G. Smith (Queen's University)

Title: Minimal free resolutions

Abstract: With the year-long objective of understanding the syzygies of generic canonical curves, we begin with a gentle introduction to minimal free resolutions of homogeneous ideals in a standard graded polynomial ring.

Curves Seminar - Mike Roth (Queen's University)

Tuesday, November 27th, 2018

Time: 2:00-3:30 p.m Place: Jeffery Hall 116

Speaker: Mike Roth (Queen's University)

Title: Beauville’s results on classification of Kähler manifolds with $c_1(K_X)=0$, II.

Abstract: We will continue the discussion of Beauville’s paper, focussing on the structure theorem for simply connected Kähler manifolds with $c_1(K_X)=0$.

Curves Seminar - Mike Roth (Queen's University)

Tuesday, November 20th, 2018

Time: 2:00-3:30 p.m Place: Jeffery Hall 116

Speaker: Mike Roth (Queen's University)

Title: Beauville’s results on classification of Kähler manifolds with $c_1(K_X)=0$.

Abstract: We will discuss Beauville’s paper giving a classification statement for (compact, complex) Kähler manifolds with vanishing first Chern class. The Hilbert scheme of points comes in to construct examples of ``irreducible holomorphic symplectic manifolds’’, one of the pieces appearing in the classification.

Curves Seminar - Mike Roth (Queen's University)

Tuesday, November 6th, 2018

Time: 2:00-3:30 p.m Place: Jeffery Hall 116

Speaker: Mike Roth (Queen's University)

Title: Local description of the Hilbert scheme when $n=2$, II.

Abstract: We will finish the discussion of local descriptions of the quotient by a finite group, and use them to give local pictures of $X^{[n]}$, $\operatorname{Sym}^n(X)$, the universal family over $X^{[n]}$, and the maps between them when $X$ is a smooth surface and $n=2$.

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