Twisted Bundles and Admissible Covers

By D. Abramovich, Alessio Corti, Angelo Vistoli. 2001

397 citations recorded by the source corpus.

Abstract We study the structure of the stacks of twisted stable maps to the classifying stack of a finite group G—which we call the stack of twisted G-covers, or twisted G-bundles. For a suitable group Gwe show that the substack corresponding to admissible G-covers is a smooth projective fine moduli space. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/93a5891d1abf5236fac9bcdf6a0bdbd8e745d3a6

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Burniat surfaces I: fundamental groups and moduli of primary Burniat surfaces (2009, background): “f polarizations should ignore the Gm-action on the line bundles. The issue is treated extensively in the literature. A procedure for removing the redundant action, called rigidiﬁcation, is treated in [ACV03], [Rom05], [AGV08], [AOV08a, Appendix A]. This is foreshadowed by the appendix in [Art74]. Go”
Gromov-Witten theory of Deligne-Mumford stacks (2006, methodology): “We recall the concept of rigidification of an algebraic stack, as presented in [ ACV ], see also the related treatment in [Ro].”
Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields (2009, neutral): “H (Xs̄,K)→ H (Xη̄,K)→ H (Xs̄,RΦη̄K) [1] →,”
Twisted stable maps to tame Artin stacks (2008, background): “Using this notion we can then consider certain twisted stable maps C →M, from n-marked genus g twisted curves C to M (see section 4 for precise definitions), and obtain a stack Kg,n(M) classifying twisted stable maps to M.”
Moduli of complexes on a proper morphism (2005, background): “Since the inertia stack of sD pug is naturally identified with Gm, this follows immediately from the theory of rigidification of Abramovich-Corti-Vistoli [2], or (in the case of simple sheaves) from an argument of Artin (contained in the appendix to [5]).”
Higher genus Gromov–Witten invariants as genus zero invariants of symmetric products (2003, methodology): “The deformations of an étale cover C′ → C are the same as deformations of the base, as in [1]; therefore the map Mη → Mt(η) is étale.”
Classical and minimal models of the moduli space of curves of genus two (2004, background): “. In particular, ξbis not normal crossings. Proposition 3.5 (Ramiﬁcation formula for Q: M 2 →M 2) KM 2 +αδ≡Q∗(K M2 +α∆ 0 + 1+α 2 ∆ 1 + 1 2 Ξ) proof: Lemma 3.3 allows us to consider the rigidiﬁcation ([ACV] §5) of M 2 with respect to the group hιigenerated by the canonical involution r: M 2 →M <ι&”
Using stacks to impose tangency conditions on curves (2003, background): “The local description for C given in [ ACV , 2.1] says that at a marking Σi, C looks ´ etale locally like the quotient of A 1 by the cyclic action of µri which fixes the origin, for some integer ri which is”
Period and index in the Brauer group of an arithmetic surface (2007, background): “If we represent A by an element A of order l in PGLl, then the regularity implies that any lift A of A in GLl ⊗Lsep such that A = 1 gives the regular representation of Z/lZ on (Lsep)l, and thus has trivial determinant if l is odd, or determinant [1] if l = 2.”
Néron models and compactified Picard schemes over the moduli stack of stable curves (2005, methodology): “s developed in recent years, in fact P d,g and Pd g are geometric GITquotients hence our stacks are “quotient stacks”, which have been carefully studied by many authors. In particular, we use [AV01], [ACV01], [E00], [LM] and [Vi89], together with the seminal paper [DM69]. Why should P d,gbe a good c”
Logarithmic Stable Maps (2008, background): “HomDb(M)(Rπ∗f LUetw/Betw ⊗ ωπ[1], L • M/MB) since πLM/MB⊗ ωπ[1] ∼= πLM/MB and Rπ! is left adjoint to π !.”
Orbifold quantum Riemann–Roch, Lefschetz and Serre (2005, background): “nceptual meaning of this formalism, one can work with the formalism without knowing this loop space interpretation. Consider the stack LX= Map(S1,X) of loops in X. We identify S1 with the interval I= [0,1] where the end points are glued together. In this way a loop is a map γ: I→Xtogether with an el”