Riemann-Roch for Deligne-Mumford stacks

We give a simple proof of the Riemann-Roch theorem for Deligne-Mumford stacks using the equivariant Riemann-Roch theorem and the localization theorem in equivariant K-theory together with some basic commutative algebra of Artin rings.Comment: 30 pages, to appear in proceedings of the Harris 60 conferenc

Notes on the construction of the moduli space of curves

The purpose of these notes is to give an introduction to Deligne-Mumford stacks and their moduli spaces, with emphasis on the moduli problem for curves. The paper has 4 sections. In section 1 we discuss the general problem of constructing a moduli "space" of curves. In section 2 we give an introduction to Deligne-Mumford stacks and their moduli spaces. In section 3 we return to curves and outline Deligne and Mumford's proof that the stack of stable curves is a smooth and irreducible Deligne-Mumford stack which is proper over Spec Z. In section 4 we explain how to use geometric invariant to construct moduli spaces for quotient stacks. Finally, we briefly outline Gieseker's geometric invariant theory construction of the moduli scheme of projective curves defined over an algebraically closed field. These notes are a slightly revised version of notes which the author has circulated privately for several years, and are based on lectures the author gave at the Weizmann Institute in July 1994. They are also available at the URL http://math.missouri.edu/~edidin/Papers/ Any updates will be posted to this URL.Comment: Latex2e 23 page

Strong regular embeddings of Deligne-Mumford stacks and hypertoric geometry

We introduce the notion of strong regular embeddings of Deligne-Mumford stacks. These morphisms naturally arise in the related contexts of generalized Euler sequences and hypertoric geometry.Comment: 25 page

Projections and Phase retrieval

We characterize collections of orthogonal projections for which it is possible to reconstruct a vector from the magnitudes of the corresponding projections. As a result we are able to show that in an $M$-dimensional real vector space a vector can be reconstructed from the magnitudes of its projections onto a generic collection of $N \geq 2M-1$ subspaces. We also show that this bound is sharp when $N = 2^k +1$. The results of this paper answer a number of questions raised in \cite{CCPW:13}.Comment: 10 page

Equivariant geometry and the cohomology of the moduli space of curves

In this expository article we give a categorical definition of the integral cohomology ring of a stack. We show that for quotient stacks the categorical cohomology may be identified with equivariant cohomology. Via this identification we show that for Deligne-Mumford quotient stacks this cohomology is rationally isomorphic to the rational cohomology of the coarse moduli space. The theory is presented with a focus on the stacks of smooth and stable curves.Comment: 34 pages - to appear in forthcoming Handbook of Moduli, edited by G. Farkas and I. Morriso

Compactifying normal algebraic spaces

The author wrote this note after being asked about the existence of compactifications of algebraic spaces. Subsequent to posting the article to the math arXiv, the author learned from Yutakaa Matsuura that the results of this paper had been proved by Raoult in 1971, using the same techniques. Since Raoult's article may be unknown to those working in the field, the author is keeping this preprint on the arXiv server. However, he makes no claim of originality.Comment: The author has been informed that the results in this paper were proved by Raoult in 1971. The preprint will remain on the server, but the author makes no claim of originalit

Nonabelian localization in equivariant K-theory and Riemann-Roch for quotients

We prove a localization formula in equivariant algebraic $K$-theory for an arbitrary complex algebraic group acting with finite stabilizer on a smooth algebraic space. This extends to non-diagonalizable groups the localization formulas H.A. Nielsen in equivariant $K$-theory of vector bundles and R.W. Thomason for higher $K$-theory of equivariant coherent sheaves. As an application we give a Riemann-Roch formula for quotients of smooth algebraic spaces by proper group actions. This formula extends previous work of B. Toen for stacks with quasi-projective moduli spaces and the authors for quotients by diagonalizable groups.Comment: To appear in Advances in Math (Artin volume); 38pages, latex.Minor revisions and deletions from previous versio

Algebraic Cuts

Let $X$ be a projective variety with a torus action, which for simplicity we assume to have dimension 1. If $X$ is a smooth complex variety, then the geometric invariant theory quotient $X//G$ can be identifed with the symplectic reduction $X_r$. Lerman introduced a construction (valid for symplectic manifolds) called symplectic cutting, which constructs a manifold $X_c$, such that $X_c$ is the union of $X_r$ and an open subset $X_{>0} \subset X$. Moreover, there is a natural torus action on $X_c$ such that $X_r$ is a component of the fixed locus. Using localization for equivariant cohomology, this construction can be used to study of $X_r$. In this note, we give an algebraic version of this construction valid for projective but possibly singular varieties defined over arbitrary fields. This construction is useful for studying $X_r$ from the point of view of algebraic geometry, using the equivariant intersection theory developed by the authors. At the end of the paper we briefly give an adaptation of Lerman's proof of the Kalkman residue formula and use it to give some formulas for characteristic numbers of quotients by a torus.Comment: Latex2e with amssymb package, 12 page

The integral Chow ring of the stack of at most 1-nodal rational curves

We give a presentation for the stack of rational curves with at most 1 node as the quotient by GL(3) of an open set in a 6-dimensional irreducible representation. We then use equivariant intersection theory to calculate the integral Chow ring of this stack.Comment: 14 pages, Latex2e, email for Dan Edidin is [email protected]

The Gromov-Witten and Donaldson-Thomas correspondence for trivial elliptic fibrations

We study the Gromov-Witten and Donaldson-Thomas correspondence conjectured in [MNOP1, MNOP2], for trivial elliptic fibrations. In particular, we verify the Gromov-Witten and Donaldson-Thomas correspondence for primary fields when the threefold is $E \times S$ where $E$ is a smooth elliptic curve and $S$ is a smooth surface with numerically trivial canonical class.Comment: 17 page