Martin Olsson

Sheaves on Artin stacks

Abstract We develop a theory of quasi-coherent and constructible sheaves on algebraic stacks correcting a mistake in the recent book of Laumon and Moret-Bailly. We study basic cohomological properties of such sheaves, and prove stack-theoretic versions of Grothendiecks Fundamental Theorem for proper morphisms, Grothendiecks Existence Theorem, Zariskis Connectedness Theorem, as well as finiteness theorems for proper push forwards of coherent and constructible sheaves. We also explain how to define a derived pullback functor which enables one to carry through the construction of a cotangent complex for a morphism of algebraic stacks due to Laumon and Moret-Bailly.

Hom-stacks and restriction of scalars

Fix an algebraic space S, and let X and Y be separated Artin stacks of finite presentation over S with finite diagonals (over S). We define a stack HomS(X ,Y) classifying morphisms between X and Y. Assume that X is proper and flat over S, and fppf–locally on S there exists a finite finitely presented flat cover Z → X with Z an algebraic space. Then we show that HomS(X ,Y) is an Artin stack with quasi–compact and separated diagonal. 1. Statements of results Fix an algebraic space S, let X and Y be separated Artin stacks of finite presentation over S with finite diagonals. Define HomS(X ,Y) to be the fibered category over the category of S–schemes, which to any T → S associates the groupoid of functors XT → YT over T , where XT (resp. YT ) denotes X ×S T (resp. Y ×S T ). Theorem 1.1. Let X and Y be finitely presented separated Artin stacks over S with finite diagonals. Assume in addition that X is flat and proper over S, and that locally in the fppf topology on S there exists a finite and finitely presented flat surjection Z → X from an algebraic space Z. Then the fibered category HomS(X ,Y) is an Artin stack locally of finite presentation over S with separated and quasi–compact diagonal. If Y is a Deligne– Mumford stack (resp. algebraic space) then HomS(X ,Y) is also a Deligne–Mumford stack (resp. algebraic space). Remark 1.2. If S is the spectrum of a field and X is a Deligne–Mumford stack which is a global quotient stack and has quasi–projective coarse moduli space, then by ([15], 2.1) there always exists a finite flat cover Z → X . Remark 1.3. When S is arbitrary and X is a twisted curve in the sense of ([1]), it is also true that etale locally on S there exists a finite flat cover Z → X as in (1.1). This is shown in ([19]). Remark 1.4. It is possible to prove that HomS(X ,Y) is an Artin stack under weaker assumptions on the diagonals of X and Y (though for the diagonal of HomS(X ,Y) to have reasonable properties the assumptions of (1.1) seem necessary). This has recently been shown by Aoki. Theorem 1.1 will be deduced from another result about pushforwards of stacks. Let S/S be a separated Artin stack locally of finite presentation and with finite diagonal. For any morphism of algebraic spaces f : S → T , define f∗S to be the fibered category over T which to any T ′/T associates the groupoid S(T ′ ×T S), with the natural notion of pullback. We call f∗S the restriction of scalars of S from S to T . Theorem 1.5. Let f : S → T be a proper, finitely presented, and flat morphism of algebraic spaces. Then the fibered category f∗S is an Artin stack locally of finite presentation over T 1

Quot Functors for Deligne-Mumford Stacks

Abstract Given a separated and locally finitely-presented Deligne-Mumford stack 𝒳 over an algebraic space S, and a locally finitely-presented 𝒪𝒳-module ℱ, we prove that the Quot functor Quot(ℱ/𝒳/S) is represented by a separated and locally finitely-presented algebraic space over S. Under additional hypotheses, we prove that the connected components of Quot(ℱ/𝒳/S) are quasi-projective over S. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Log) twisted curves

We describe an equivalence between the notion of balanced twisted curve introduced by Abramovich and Vistoli, and a new notion of log twisted curve, which is a nodal curve equipped with some logarithmic data in the sense of Fontaine and Illusie. As applications of this equivalence, we construct a universal balanced twisted curve, prove that a balanced twisted curve over a general base scheme admits étale locally on the base a finite flat cover by a scheme, and also give a new construction of the moduli space of stable maps into a Deligne–Mumford stack and a new proof that it is bounded.

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 Kontsevichs 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.

Compactifying moduli spaces for Abelian varieties

A Brief Primer on Algebraic Stacks.- Preliminaries.- Moduli of Broken Toric Varieties.- Moduli of Principally Polarized Abelian Varieties.- Moduli of Abelian Varieties with Higher Degree Polarizations.- Level Structure.

Algebraic spaces and stacks

geometrie algebrique # espace algebrique # empilement algebrique # fibration # espace de modules fins # espace de modules grossiers

Nagata compactification for algebraic spaces

We prove the Nagata compactification theorem for any separated map of finite type between quasi-compact and quasi-separated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian approximation for such algebraic spaces, generalizing earlier results in the case of schemes.

Semistable degenerations and period spaces for polarized K3 surfaces

Modular compactifications of moduli spaces for polarized K3 surfaces are constructed using the tools of logarithmic geometry in the sense of Fontaine and Illusie. The relationship between these new moduli spaces and the classical minimal and toroidal compactifications of period spaces are discussed, and it is explained how the techniques of this paper yield models for the latter spaces over number fields. The paper also contains a discussion of Picard functors for log schemes and a logarithmic version of Artin’s method for proving representability by an algebraic stack.

On the global quotient structure of the space of twisted stable maps to a quotient stack

Let 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 Kg,n(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 Kg,n(X , d) using Hilbert schemes and admissible G-covers.