I. PREPARATION

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

Venue: C2

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

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

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

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

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

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

Relations in Pic(M_g) (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

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 M_0,n). Problem Session: Han-bom Moon.

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

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.

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 M_g). 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.

Gavril Farkas Ideals of canonical curves and the canonical model of M_g: (Koszul cohomology, resolutions of ideals of canonical curves. Keel’s pipedream of the canonical model of M_g). 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 M_0,n: (Weighted pointed curves. GIT constructions of Veronese models of M_0,n. Log minimal models for M_0,n). Problem Session: Ian Morrison.

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 M_0,n: (Examples of Chen and Möller. Hypergraph examples. Two conics examples). Problem Session: Ana-Maria Castravet.