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, algebraic stacks, invariant theory, algebraic curves, higher-dimensional geometry, minimal model program, Brauer groups, and singularity theory.

CONFERENCES

I'm co-organizing a conference on "Moduli spaces and moduli stacks" at Columbia University from May 23-27, 2011. The webpage is here.

TEACHING

• Algebraic Number Theory (Fall 2010)
• Calculus III (Fall 2010)
• Calculus III (Fall 2008)

PUBLICATIONS

• Good moduli spaces for Artin stacks, pdf.
  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 .
  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 .
  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.

• Computing invariants via slicing: Gel'fand MacPherson, Gale and positive characteristic Kontsevich, 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.

• Adequate moduli spaces and geometrically reductive group schemes, pdf.
  We develop the analogue of the theory of good moduli spaces in characteristic p (and mixed characteristic) characterizing quotients by geometrically reductive group schemes.

• Recasting Results in Equivariant Geometry: Affine Cosets, Observable Subgroups and Existence of Good Quotients, with Rob Easton, pdf.
  Using the language of stacks, we recast and generalize a selection of results in equivariant geometry.

• Singularities with G_m-action and the log minimal model program for $\bar{M}_g$, with Maksym Fedorchuk and David Smyth, pdf.
  We give a precise formulation of the modularity principle for the log canonical models of $\bar{M}_g$. Assuming the modularity principle holds, we develop and compare two methods for determining the critical $\alpha$-values at which a singularity or complete curve with G_m-action arises in the modular interpretations of log canonical models of $\bar{M}_g$. The first method involves a new invariant of curve singularities with G_m-action, constructed via the characters of the induced G_m-action on spaces of pluricanonical forms. The second method involves intersection theory on the variety of stable limits of a singular curve. We compute the expected $\alpha$-values for large classes of singular curves, including curves with ADE, toric and monomial unibranch Gorenstein singularities as well as for ribbons, and show that the two methods yield identical predictions. We use these results to give a conjectural outline of the log MMP for $\bar{M}_g$.

• Invariant rings through categories, with Johan de Jong, pdf.
  We formulate a notion of "geometric reductivity" in an abstract categorical setting which we refer to as adequacy. The main theorem states that the adequacy condition implies that the ring of invariants is finitely generated. This result applies to the category of modules over a bialgebra, the category of comodules over a bialgebra, and the category of quasi-coherent sheaves on a finite type algebraic stack over an affine base.

• Weakly proper moduli stacks of curves, with David Smyth and Frederick van der Wyck, pdf.
  This is the first in a projected series of three papers in which we construct the second flip in the log minimal model program for $\bar{M}_g$. We introduce the notion of a weakly proper algebraic stack, which may be considered as an abstract characterization of those mildly non-separated moduli problems encountered in the context of Geometric Invariant Theory (GIT), and develop techniques for proving that a stack is weakly proper without the usual semistability analysis of GIT. We define a sequence of moduli stacks of curves involving nodes, cusps, tacnodes, and ramphoid cusps, and use the aforementioned techniques to show that these stacks are weakly proper. This will be the key ingredient in forthcoming work, in which we will prove that these moduli stacks have projective good moduli spaces which are log canonical models for $\bar{M}_g$.