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Martin Olsson

Publications and source records attributed to Martin Olsson.

30 records · Page 2Linked to original sources
arXiv2010

Twisted stable maps to tame Artin stacks

This paper is a continuation of our earlier development of a theory of tame Artin stacks. Our main goal here is the construction of an appropriate analogue of Kontsevich's space of stable maps in the case where the target is a tame Artin stack. When the target is a tame Deligne--Mumford stack, the theory was developed by Abramovich and Vistoli, and found a number of applications. The theory for arbitrary tame Artin stacks developed here is very similar, but it is necessary to overcome a number of technical hurdles and to generalize a few questions of foundation.

math.AG↗
arXiv2008

Quot Functors for Deligne-Mumford Stacks

Given a separated and locally finitely-presented Deligne-Mumford stack $\cX$ over an algebraic space $S$, and a locally finitely-presented $\OO_{\cX}$-module $\cF$, we prove that the Quot functor $\text{Quot}(\cF/\cX/S)$ is represented by a separated and locally finitely-presented algebraic space over $S$. Under additional hypotheses, we prove that the connected components of $\text{Quot}(\cF/\cX/S)$ are quasi-projective over $S$.

math.AG↗
arXiv2007

Generators and relations for the etale fundamental group

If $C$ is a smooth curve over an algebraically closed field $k$ of characteristic $p$, then the structure of the maximal prime to $p$ quotient of the étale fundamental group is known by analytic methods. In this paper, we discuss the properties of the fundamental group that can be deduced by purely algebraic techniques. We describe a general reduction from an arbitrary curve to the projective line minus three points, and show what can be proven unconditionally about the maximal pro-nilpotent and pro-solvable quotients of the prime-to-$p$ fundamental group. Included is an appendix which treats the tame fundamental group from a stack-theoretic perspective.

math.AG↗
arXiv2007

The Picard group of $M_{1,1}$

We compute the Picard group of the moduli stack of elliptic curves and its canonical compactification over general base schemes.

math.AG↗
arXiv2007

Tame stacks in positive characteristic

We introduce and study a class of algebraic stacks with finite inertia in positive and mixed characteristic, which we call tame algebraic stacks. They include tame Deligne-Mumford stacks, and are arguably better behaved than general Deligne-Mumford stacks. We also give a complete characterization of finite flat linearly reductive schemes over an arbitrary base. Our main result is that tame algebraic stacks are étale locally quotient by actions of linearly reductive finite group schemes. In a subsequent paper we will show that tame algebraic stacks admit a good theory of stable maps.

math.AG↗
arXiv2006

Perverse sheaves on Artin stacks

In this paper we develop the theory of perverse sheaves on Artin stacks continuing the study in "The six operations for sheaves on Artin stacks I: Finite Coefficients" and "The six operations for sheaves on Artin stacks II: Adic Coefficients" (math.AG/0512097 and math.AG/0603680)

math.AG↗
arXiv2006

On The Global Quotient Structure of The Space of Twisted Stable Maps to a Quotient Stack

Let $\mathcal{X}$ be a tame proper Deligne-Mumford stack of the form $[M/G]$ where $M$ is a scheme and $G$ is an algebraic group. We prove that the stack $\mathcal{K}_{g,n}(\mathcal{X},d)$ of twisted stable maps is a quotient stack and can be embedded into a smooth Deligne-Mumford stack. When $G$ is finite, we give a more precise construction of $\mathcal{K}_{g,n}(\mathcal{X},d)$ using Hilbert schemes and admissible $G$-covers.

math.AG↗
arXiv2002

Kawamata-Viehweg vanishing as Kodaira vanishing for stacks

We associate to a pair $(X,D)$, consisting of a smooth scheme with a divisor $D\in \text{Div}(X)\otimes \mathbb{Q}$ whose support is a divisor with normal crossings, a canonical Deligne--Mumford stack over $X$ on which $D$ becomes integral. We then reinterpret the Kawamata--Viehweg vanishing theorem as Kodaira vanishing for stacks.

math.AG↗
arXiv2002

Miracles and complementarity in de Sitter space

In this paper we consider a scenario, consisting of a de Sitter phase followed by a phase described by a scale factor $a(t)\sim t^{q}$, where $1/3<q<1$, which can be viewed as an inflationary toy model. It is argued that this scenario naively could lead to an information paradox. We propose that the phenomenon of Poincaré recurrences plays a crucial role in the resolution of the paradox. We also comment on the relevance of these results to inflation and the CMBR.

hep-th↗