Department of Mathematics Stanford University 450 Serra Mall, Building 380 Stanford, CA 94305 Email: jarod at math dot stanford dot edu Phone: (415) 572-7820

RESEARCH INTERESTS

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

PUBLICATIONS

• Good moduli spaces for Artin stacks, pdf (updated July 10, 2008).
  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.

• Local properties of Artin stacks and their moduli spaces, pdf (updated April 14, 2008).
  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 are formally local quotients stacks by the stabilizer and admit formally locally good moduli spaces. We conjecture that these statements hold etale-locally. We use this theory to show that good moduli spaces are universal for maps to algebraic spaces. We also give a stack-theoretic proof and generalization of Luna's slice theorem.

• From Artin stacks to classical invariant theory: Gel'fand-MacPherson, Gale, and positive characteristic Kontsevich, in preparation.
  We offer a stack-theoretic approach to computing invariants. We give generalizations of the Gel'fand-MacPherson correspondence and the Gale transform, which incorporate similar generalizations by Borcea and Hu. The use of stacks offers immediate geometric consequences. We give Zariski-local descriptions of moduli spaces of ordered configurations of linear subspaces. Finally, we offer an explicit description of the moduli space M_0(P^1,2) in characteristic 2. The singularity at the totally ramified cover is isomorphic to the affine cone over the Veronese embedding P^1 --> P^4 which is not a Q-Gorenstein (hence not finite quotient) singularity.