Research

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I'm interested in algebraic geometry and representation theory. Most of my work involves the use of algebraic stacks and English; I like to think I have a decent command of both, but I have more new ideas about the former. To get a better feel for the sort of stuff I like to think about, check out my MathOverflow user page.


Publications

Me on the arXiv.

A "bottom up" characterization of smooth Deligne-Mumford stacks with Matthew Satriano. arXiv:1503.05478
There is no degree map for 0-cycles on Artin stacks with Dan Edidin and Matthew Satriano. arXiv:1208.3239 Transformation Groups June 2013
Formal GAGA for good moduli spaces with David Zureick-Brown. arXiv:1208.2882 Algebraic Geometry May 2015
Torus Quotients as Global Quotients by Finite Groups with Matthew Satriano. arXiv:1201.4807 JLMS 2015
Toric Stacks I: The Theory of Stacky Fans with Matthew Satriano. arXiv:1107.1906 Transactions of the AMS 2015
Toric Stacks II: Intrinsic Characterization of Toric Stacks with Matthew Satriano. arXiv:1107.1907 Transactions of the AMS 2015

Toric Stacks

Because of the rich dictionary between combinatorics and geometry, toric varieties are an extremely fertile testing ground for ideas in algebraic geometry. Once well versed in this dictionary, one can often cook up examples tailored to exhibit some desired behavior, and then understand just about everything about those examples. Since stacks are often difficult to handle and think about, it is desirable to have a similarly versatile-yet-easy-to-handle class of stacks.

The following two papers aim at a grand unified theory of toric stacks (there are at least seven distinct definitions in the literature prior to this). The first paper develops the combinatorics-geometry dictionary between stacky fans and toric stacks, focusing on stacky phenomena, like canonical stacks, good moduli space morphisms, and moduli interpretations. The second paper is dedicated towards an intrinsic characterization of toric stacks, identifying them as normal reduced finite type stacks with a dense open torus satisfying some reasonable hypotheses (e.g. affine diagonal, linearly reductive stabilizers).

  • Toric Stacks I: The Theory of Stacky Fans with Matthew Satriano. arXiv:1107.1906
  • Toric Stacks II: Intrinsic Characterization of Toric Stacks with Matthew Satriano. arXiv:1107.1907

Torus quotients as quotients by finite groups

Suppose $G$ is a finite group acting on a smooth scheme $U$. The singularities of the quotient scheme $X=U/G$ are quotient singularities. William Fulton posed the question, "is every scheme with tame quotient singularities globally a quotient of a smooth scheme by the action of a finite group?" The goal is to find a stack $\X$ with coarse space $X$, and to endow $\X$ with a global quotient structure. Though this reformulation is trivial, it has major advantages. First, if $\X\cong [U/G]$, then $U$ is automatically smooth. Second, quotient structures on $\X$ are witnessed by vector bundles. $X$ is automatically the coarse space of a smooth DM stack, which is automatically a quotient by some (infinite) algebraic group. This gives a smooth DM stack with a special vector bundle on it. The goal is then to modify the stack (without changing the coarse space) and the vector bundle to eventually get a smooth stack with a vector bundle that witnesses it as a finite quotient.

The paper below has two main results:

  1. Any quasi-projective quotient of a smooth space by a split torus (acting with finite stabilizers) is a quotient of a smooth space by a finite diagonalizable group. In particular, this gives you a procedure for expressing quasi-projective simplicial toric varieties as quotients of smooth varieties by finite groups (note: such varieties cannot be expressed as quotients of smooth toric varieties in general, so this application is non-trivial).
  2. A space with quotient singularities is a torus quotient (as above) if and only if its canonical stack is a the stack quotient of a smooth space by a torus. Alternatively, a variety is a torus quotient if and only if there are Weil divisors which generate the class groups of the completed local rings (moreover, these local class groups are fairly easy to compute). Note that the second version is clearly Zariski local.

Using the second result, we show that the projective surface $\PP^2/A_5$ cannot be expressed as a quotient of a smooth variety by a finite abelian group, even if you resolve two of its three singularities.