On Coverings of Deligne–mumford Stacks and Surjectivity of the Brauer Map

Kresch, Andrew; Vistoli, Angelo

doi:10.1112/s0024609303002728

Cited by 54 publications

(58 citation statements)

References 12 publications

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“…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.…”

Section: Statements Of Resultsmentioning

confidence: 99%

Hom-stacks and restriction of scalars

Olsson

2006

Duke Math. J.

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Abstract. 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 Hom S (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 Hom S (X , Y) is an Artin stack with quasi-compact and separated diagonal. Statements of resultsFix an algebraic space S, let X and Y be separated Artin stacks of finite presentation over S with finite diagonals. Define Hom S (X , Y) to be the fibered category over the category of S-schemes, which to any T → S associates the groupoid of functors X T → Y T over T , whereTheorem 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 Hom S (X , Y) is an Artin stack locally of finite presentation over S with separated and quasi-compact diagonal. If Y is a DeligneMumford stack (resp. algebraic space) then Hom S (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 thatétale locally on S there exists a finite flat cover Z → X as in (1

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Section: Statements Of Resultsmentioning

confidence: 99%

Hom-stacks and restriction of scalars

Olsson

2006

Duke Math. J.

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“…This is always the case for smooth orbifolds with quasi-projective moduli spaces. Another sufficient condition is that the moduli space of the stack is quasi-projective and of dimension for which the Brauer map is known to be surjective [2], [16]. From now on we will tacitly assume that all the stacks we are dealing with satisfy one of these conditions.…”

Section: Presentations As Global Quotientsmentioning

confidence: 99%

String compactifications on Calabi–Yau stacks

2006

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In this paper we study string compactifications on Deligne-Mumford stacks. The basic idea is that all such stacks have presentations to which one can associate gauged sigma models, where the group gauged need be neither finite nor effectively-acting. Such presentations are not unique, and lead to physically distinct gauged sigma models; stacks classify universality classes of gauged sigma models, not gauged sigma models themselves. We begin by defining and justifying a notion of "Calabi-Yau stack," recall how one defines sigma models on (presentations of) stacks, and calculate of physical properties of such sigma models, such as closed and open string spectra. We describe how the boundary states in the open string B model on a Calabi-Yau stack are counted by derived categories of coherent sheaves on the stack. Along the way, we describe numerous tests that IR physics is presentationindependent, justifying the claim that stacks classify universality classes. String orbifolds are one special case of these compactifications, a subject which has proven controversial in the past; however we resolve the objections to this description of which we are aware. In particular, we discuss the apparent mismatch between stack moduli and physical moduli, and how that discrepancy is resolved. January 20051

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“…be a twisted curve and choose a projective 1-dimensional varietyC with a degree l C finite flat morphismC → C (see [24]). By pullback, there is a natural morphism Hom(C, X) → Hom(C, X).…”

Section: 42mentioning

confidence: 99%