Program

I. PREPARATION

Week 4 (23/12-27/12/2013)

Venue: C2

Tuesday

Thursday

8:30 – 10:00

Lecture

Lecture

10:00 – 10:30

Break

Break

10:30 – 11:30

Problem Session

Problem Session


Proposed schedule for Jack Huizenga’s lectures (December 3 – January 3)

A. Parameter Spaces
(3 lectures)

Moduli spaces (Functor of points. Fine moduli spaces. Coarse moduli spaces. Why is Mg not fine?)

Parameter spaces (Grassmannians. Hilbert schemes. Examples. Tautological classes.)

Representability of the Hilbert functor (Regularity. Boundedness. Pathologies.)

B. Moduli Spaces: basic properties (5 lectures)

Mg (Small genus examples. Hurwitz schemes. Dimension and irreducibility. Hodge bundle and λ. Pic(Mg). Sketch of GIT construction.)

Mg (Why we need to compactify Mg. Semistable reduction. Deformations of nodal curves. Basic properties of stable curves. Components of ∆ and Pic(Mg).)

Mg,n (Pointed curves. Gluing and forgetful maps. ψ classes. Components of ∆ and Pic(Mg,n). Pullback formulae. Dual graphs and stratification by topological type.)

C. Working with divisors on moduli Spaces (4 lectures)

Relations in Pic(Mg) (Test curves. Key examples. Grothendieck-Riemann-Roch. Mumford’s formula. Canonical bundle.)

Ample classes (Ample classes provided by GIT constructions. Elliptic tails. Cornalba-Harris Theorem. The F-conjecture.)

Effective classes (Brill-Noether theory. Eisenbud-Harris-Mumford theorem. Effective cones and the slope conjecture).

 

II. WORKSHOP

Proposed schedule week of January 6-10


Joe Harris
 Parameter spaces: (Roster of spaces that parameterize curves. Basic questions about these spaces. Our current knowledge (and ignorance) about these questions). Problem Session: Izzet Coskun.


Dawei Chen
 Effective classes: (Constructions of moving and extremal effective curves. Applications and connections to Teichmüller dynamics). Problem Session: Anand Deopurkar.


Maksym Fedorchuk
 Positivity and projectivity: (Intrinsic approaches to positivity of divisor classes and projectivity of moduli spaces. Examples on M0,n). Problem Session: Han-bom Moon.

Tuesday, January 7

Thursday, January 9

8:45

Joe Harris (Part 1)

Joe Harris (Part 2)

9:30

Dawei Chen

Maksym Fedorchuk

10:30

Tea

Tea

11:00

Problem session

Problem session



Proposed schedule week of January 13-17


Jarod Alper
 Introduction to the log minimal model program: (Contractions, flips and flops. What is a log model? Setting up the program for Mg). Problem Session: David Smyth.


David Smyth
 Intrinsic constructions of log minimal models: (Predictions using the modularity conjecture. Minimal models as stacks. Using positivity to construct good moduli spaces.) Problem Session: Jarod Alper.


Matthew Woolf
 Introduction to derived Categories: (Category of modules over a finite dimensional algebra and quivers. Abelian categories. Derived categories. Beilinson’s equivalence for projective space). Problem Session: Arend Bayer.

Tuesday, January 14

Thursday, January 16

8:45

Matthew Woolf (Part 1)

Matthew Woolf (Part 2)

9:30

Jarod Alper

David Smyth

10:30

Tea

Tea

11:00

Problem session

Problem session


Proposed schedule week of January 20-24


Yongnam Lee
 GIT constructions of log minimal models: (Predictions of critical values and exceptional loci using the F-conjecture. Matching predictions to GIT quotients. The Hassett-Hyeon plan. The first contraction.) Problem Session: Nicola Tarasca.


Emanuele Macri
 Bridgeland stability conditions: (Slope stability for curves. Bridgeland’s definition. Bridgeland’s deformation theorem. Moduli spaces of stable objects.) Problem Session: Matthew Woolf.


Arend Bayer
 Variation of stability.: (King and Bridgeland stability for modules. Moduli spaces of quiver representations. Examples for the projective plane). Problem Session: Emanuele Macri.

Tuesday, January 21

Thursday, January 23

8:45

Yongnam Lee (Part 1)

Yongnam Lee (Part 2)

9:30

Emanuele Macri

Arend Bayer

10:30

Tea

Tea

11:00

Problem session

Problem session



Proposed schedule week of February 10-14


David Hyeon
 The log minimal model program for M3: (GIT constructions of genus 3 log minimal models. Relations to other moduli spaces). Problem Session: Ian Morrison.

Ian Morrison GIT of pluricanonical curves: (The Hilbert-Mumford criterion for Hilbert points. Asymptotic stability of smooth curves. The Gieseker-Mumford construction of Mg). Problem Session: Filippo Viviani.

Filippo Viviani GIT for polarized curves: (Caporaso’s results. Threshholds as the ratio [d/(g−1)] decreases. Examples involving elliptic tails). Problem Session: Ian Morrison.

Tuesday, February 11

Thursday, February 13

8:45

Ian Morrison (Part 1)

Ian Morrison (Part 2)

9:30

Filippo Viviani

David Hyeon

10:30

Tea

Tea

11:00

Problem session

Problem session



Proposed schedule week of February 17-21


Gavril Farkas
 Ideals of canonical curves and the canonical model of Mg: (Koszul cohomology, resolutions of ideals of canonical curves. Keel’s pipedream of the canonical model of Mg). Problem Session: Angela Ortega.

Angela Ortega Log canonical models of curves of genus 2: (GIT of binary sextics and the log minimal model program for M2. Prym varieties and spin curves with a focus on genus 2). Problem Session: Gavril Farkas.

Angela Gibney GIT and log minimal models for M0,n: (Weighted pointed curves. GIT constructions of Veronese models of M0,n. Log minimal models for M0,n). Problem Session: Ian Morrison.

Tuesday, February 18

Thursday, February 20

8:45

Angela Gibney (Part 1)

Angela Gibney (Part 2)

9:30

Angela Ortega

Gavril Farkas

10:30

Tea

Tea

11:00

Problem session

Problem session



Proposed schedule week of February 24-28


Ana-Maria Castravet
 Extremal effective curves in small genus: (Hypergraph curves in genus 0. Non-hypergraph examples of Opie and of Giansiracusa and Doran. Examples of Chen and Coskun in genus 1). Problem Session: Jenia Tevelev.


Jenia Tevelev
 Rigid curves on M0,n: (Examples of Chen and Möller. Hypergraph examples. Two conics examples). Problem Session: Ana-Maria Castravet.

Tuesday, February 25

Thursday, February 27

9:30

Ana-Maria Castravet

Jenia Tevelev

10:30

Tea

Tea

11:00

Problem session

Problem session