Geometry at the ANU: Graduate student workshop

Geometry at the ANU: Conference

Australian National University, Canberra, Australia

August 15-19, 2016

ABSTRACTS

The workshop will consist of the following lecture series:

  • Ana-Maria Castravet (Northeastern), Birational geometry of moduli spaces of rational curves, 3 lectures
    • Abstract: I will start these lectures by discussing Hassett's moduli spaces of weighted stable rational curves and continue with an introduction to Mori Dream Spaces. The final goal will be to present a sketch of the proof that the Grothendieck-Knudsen moduli space of stable rational curves with n markings is not a Mori Dream Space when n is large, by reducing the question to a study of blow-ups of weighted projective planes.

  • François Charles (Paris-Sud), Divisors on varieties with trivial canonical bundle, 4 lectures

  • Frank Gounelas (Humboldt), Positivity of vector bundles and birational geometry, 2 lectures

    Abstract: In these talks I will introduce various notions of positivity (e.g. ampleness) for vector bundles on projective varieties and explain through examples how the birational geometry of the variety is governed by positivity of the cotangent bundle.

  • Emanuele Macri (Northeastern), Introduction to Bridgeland stability, 4 lectures

    Abstract: In this series of lectures we will cover the following topics:
    (1) Review on stability of vector bundles on curves.
    (2) Stability conditions on derived categories.
    (3) Moduli spaces of stable objects and variation of stability.
    (4) The case of surfaces.
    (5) The higher dimensional case and open problems.

  • Brett Parker (ANU), Holomorphic curves and tropical curves, 1 lecture

    Abstract: Holomorphic curves have many fascinating applications to algebraic geometry, symplectic geometry, and string theory from physics, but they can be difficult to find and visualise. I will explain how in many situations, we can study holomorphic curves by studying piecewise linear graphs called tropical curves.

  • Burt Totaro (UCLA), Algebraic cycles and birational geometry, 4 lectures

    Abstract: The course is centered on the Chow group of algebraic cycles on an algebraic variety. Some introductions to this material are Hartshorne's section II.6 on Divisors and Appendix A on Intersection Theory. We will mostly use what we need from Fulton's Intersection Theory without proof.

    One main theme of the course is the notion of "decomposition of the diagonal", which describes the geometric consequences that hold for a variety with "small" Chow groups. The basic argument by Bloch and Srinivas is very simple, once you know the formal properties of Chow groups.

    The course will end with applications of these ideas to birational geometry. In particular, we will use Chow groups to prove a striking recent result: many Fano hypersurfaces X in projective space are not stably rational. (That is, no product of X with projective space is birational to projective space.)

    • PROGRAM

    MondayAugust 15 Hanna Neumann Seminar Room (G058
    9-9:30 Welcome and registration in lobby of John Dedman
    9:30-10:30 Totaro Lecture 1
    10:30-11 Coffee
    11-12 Charles Lecture 1
    12-2 Lunch break
    2-3 Macri Lecture 1
    3-3:30 Coffee
    3:30-4:30 Parker Lecture 1
     
    TuesdayAugust 16 Hanna Neumann Seminar Room (G058)
    9:30-10:30 Totaro Lecture 2
    10:30-11 Coffee
    11-12 Charles Lecture 2
    12-2 Lunch break
    2-3 Macri Lecture 2
    3-3:30 Coffee
    3:30-4:30 Castravet Lecture 1
     
    WednesdayAugust 17 Hanna Neumann Seminar Room (G058)
    9-10Macri Lecture 3
    10-10:30 Coffee
    10:30-11:30 Charles Lecture 3
    11:45-12:45 Castravet Lecture 2
    Free afternoon
     
    ThursdayAugust 18 Hanna Neumann Seminar Room (G058)
    9:30-10:30 Totaro Lecture 3
    10:30-11 Coffee
    11-12 Castravet Lecture 3
    12-2 Lunch break
    2-3 Macri Lecture 4
    3-3:30 Coffee
    3:30-4:30 Gounelas Lecture 1
     
    FridayAugust 19 John Dedman Building, room G35
    9:30-10:30 Totaro Lecture 4
    10:30-11 Coffee
    11-12 Charles Lecture 4
    12-2 Lunch break
    2-3 Gounelas Lecture 2

    * Schedule subject to change.