Department of Mathematics Columbia University 2990 Broadway New York, NY 10027 Email: jarod at math dot columbia dot edu Office phone: (212) 854-4759 Fax: (212) 854-8962

RESEARCH INTERESTS

Algebraic geometry: moduli spaces, invariant theory and algebraic stacks.

TEACHING

• Calculus III (Fall 2008)

PUBLICATIONS

• Good moduli spaces for Artin stacks, pdf (updated November 9, 2009).
  We develop the theory of associating moduli spaces with nice geometric properties to arbitrary Artin stacks generalizing Mumford's geometric invariant theory and tame stacks.

• On the local quotient structure of Artin stacks, pdf (updated July 1, 2009).
  We show that near closed points with linearly reductive stabilizer, Artin stacks are formally locally quotient stacks by the stabilizer. We conjecture that the statement holds étale locally and we provide some evidence for this conjecture. In particular, we prove that if the stabilizer of a point is linearly reductive, the stabilizer acts algebraically on a miniversal deformation space generalizing results of Pinkham and Rim. In additional, a stack-theoretic proof of Luna's \'etale slice theorem is presented.

• Local properties of good moduli spaces, pdf (updated July 10, 2009).
  We study the local properties of Artin stacks and their good moduli spaces, if they exist. We show that near closed points with linearly reductive stabilizer, Artin stacks formally locally admit good moduli spaces. In particular, the geometric invariant theory is developed for actions of linearly reductive group schemes on formal affine schemes. We also give conditions for when the existence of good moduli spaces can be deduced from the existence of \'etale charts admitting good moduli spaces.

• The Kontsevich moduli of stable maps M_0(P^1,2) in characteristic 2, pdf
  We offer a groupoid-theoretic approach to computing invariants. We illustrate this approach by describing the Gel'fand-MacPherson correspondence and the Gale transform as well as giving Zariski-local descriptions of the moduli space of ordered points in P^1. We give an explicit description of the moduli space M_0(P^1,2) over Spec Z. In characteristic 2, there is a singularity at the totally ramified cover which is isomorphic to the affine cone over the Veronese embedding P^1 --> P^4 and which is not a Q-Gorenstein (hence not finite quotient) singularity.

• Geometrically reductive group schemes and adequate moduli spaces, in preparation.
  We develop the theory of the analogue of good moduli spaces in characteristic p characterizing quotients by geometrically reductive group schemes.