Home /  Introduction to the Theory of Algebraic Curves (UC Berkeley)

Summer Graduate School

Introduction to the Theory of Algebraic Curves (UC Berkeley) July 08, 2024 - July 19, 2024
Parent Program: --
Location: UC Berkeley
Organizers Izzet Coskun (University of Illinois, Chicago), Eric Larson (Brown University), LEAD Hannah Larson (University of California, Berkeley), Isabel Vogt (Brown University)
Lecturer(s)

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Teaching Assistants(s)

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Speaker(s)

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Description
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In the last few years, there have been extraordinary developments in many aspects of curve theory. Beginning with many examples in low genus, this summer school will introduce the participants to the background behind these developments in the following areas:

  1. moduli spaces of stable curves
  2. Brill–Noether theory
  3. the extrinsic geometry of the curves in projective space

We will also include an introduction to some open problems at the forefront of these active areas.

School Structure

There will be two one-hour lectures and two problem sessions each day. 

Prerequisites

Basic knowledge of algebraic geometry up to the level of the Riemann–Roch and Riemann–Hurwitz theorems for curves. (These theorems appear, for example, in Hartshorne’s Algebraic Geometry as Theorem IV.1.3 and Corollary IV.2.4; or in Sections 2.3 and 2.1 in Griffiths–Harris Principles of Algebraic Geometry).

Reading List

Application Procedure

For eligibility and how to apply, see the Summer Graduate Schools homepage.


 

Keywords and Mathematics Subject Classification (MSC)
Tags/Keywords
  • Moduli spaces of curves

  • canonical curves

  • linear series

  • Brill–Noether theory

  • degeneration

  • Deformation Theory

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Schedule, Notes/Handouts & Videos
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Jul 08, 2024
Monday
08:40 AM - 09:00 AM
  Introduction to SLMath
09:00 AM - 10:15 AM
  Introduction to linear series on curves
Isabel Vogt (Brown University)
10:15 AM - 10:45 AM
  Break
10:45 AM - 12:00 PM
  Introduction to moduli
Hannah Larson (University of California, Berkeley)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:30 PM
  Problem Session
03:30 PM - 04:00 PM
  Afternoon Tea
04:00 PM - 05:00 PM
  Problem Session
Jul 09, 2024
Tuesday
09:00 AM - 10:15 AM
  Examples of low genus curves in projective space
Eric Larson (Brown University)
10:15 AM - 10:45 AM
  Break
10:45 AM - 12:00 PM
  Low genus canonical curves
Izzet Coskun (University of Illinois, Chicago)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:30 PM
  Problem Session
03:30 PM - 04:00 PM
  Afternoon Tea
04:00 PM - 05:00 PM
  Problem Session
Jul 10, 2024
Wednesday
09:00 AM - 10:15 AM
  Properties of the moduli space of smooth curves
Hannah Larson (University of California, Berkeley)
10:15 AM - 10:45 AM
  Break
10:45 AM - 12:00 PM
  Introduction to Brill--Noether theory
Isabel Vogt (Brown University)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:30 PM
  Problem Session
03:30 PM - 04:00 PM
  Afternoon Tea
04:00 PM - 05:00 PM
  Problem Session
Jul 11, 2024
Thursday
09:00 AM - 10:15 AM
  Introduction to stable curves
Hannah Larson (University of California, Berkeley)
10:15 AM - 10:45 AM
  Break
10:45 AM - 12:00 PM
  The dualizing sheaf
Eric Larson (Brown University)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:30 PM
  Problem Session
03:30 PM - 04:00 PM
  Afternoon Tea
04:00 PM - 05:00 PM
  Problem Session
Jul 12, 2024
Friday
09:00 AM - 10:15 AM
  Introduction to Brill-Noether theory via degeneration
Isabel Vogt (Brown University)
10:15 AM - 10:45 AM
  Break
10:45 AM - 12:00 PM
  Introduction to deformation theory I
Izzet Coskun (University of Illinois, Chicago)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:30 PM
  Problem Session
03:30 PM - 04:00 PM
  Afternoon Tea
04:00 PM - 05:00 PM
  Problem Session
Jul 15, 2024
Monday
09:00 AM - 10:15 AM
  Stable reduction I
Hannah Larson (University of California, Berkeley)
10:15 AM - 10:45 AM
  Break
10:45 AM - 12:00 PM
  Proof of Brill-Noether nonexistence
Isabel Vogt (Brown University)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:30 PM
  Problem Session
03:30 PM - 04:00 PM
  Afternoon Tea
04:00 PM - 05:00 PM
  Problem Session
Jul 16, 2024
Tuesday
09:00 AM - 10:15 AM
  Introduction to deformation theory II
Eric Larson (Brown University)
10:15 AM - 10:45 AM
  Break
10:45 AM - 12:00 PM
  Combinatorics of the boundary of M_g-bar
Hannah Larson (University of California, Berkeley)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:30 PM
  Problem Session
03:30 PM - 04:00 PM
  Afternoon Tea
04:00 PM - 05:00 PM
  Problem Session
Jul 17, 2024
Wednesday
09:00 AM - 10:15 AM
  Introduction to limit linear series
Isabel Vogt (Brown University)
10:15 AM - 10:45 AM
  Break
10:45 AM - 12:00 PM
  Proof of Brill-Noether existence
Izzet Coskun (University of Illinois, Chicago)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  Problem Session
03:00 PM - 05:00 PM
  Visit to SLMath with Afternon Tea and Group Photo
Jul 18, 2024
Thursday
09:00 AM - 10:15 AM
  Intro to intersection theory of M_g and M_{g,n}-bar
Hannah Larson (University of California, Berkeley)
10:15 AM - 10:45 AM
  Break
10:45 AM - 12:00 PM
  Normal bundles of curves I
Izzet Coskun (University of Illinois, Chicago)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:30 PM
  Problem Session
03:30 PM - 04:00 PM
  Afternoon Tea
04:00 PM - 05:00 PM
  Problem Session
Jul 19, 2024
Friday
09:00 AM - 10:15 AM
  Global aspects of Brill-Noether theory
Isabel Vogt (Brown University)
10:15 AM - 10:45 AM
  Break
10:45 AM - 12:00 PM
  Normal bundles of curves II
Eric Larson (Brown University)
12:00 PM - 01:30 PM
  Lunch
01:30 PM - 02:00 PM
  Participant Survey
02:00 PM - 03:30 PM
  Problem Session
03:30 PM - 04:00 PM
  Afternoon Tea