Twisted stable maps to tame Artin stacks

By D. Abramovich, Martin Olsson, Angelo Vistoli. 2008

113 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d057e6849c6a24f5e6cfd52d3c412d84edabb53c

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In Geometry and Topology

Citations in the corpus

Logarithmic Stable Maps (2008, background): “� Lemma 6.1.2. ([ 3 ]) Let F be a stack. Suppose that there are: a morphism from F to an algebraic stack G and a smooth surjective map from a scheme U to G, for which F ×G U is algebraic.”
Orbifold quasimap theory (2014, background): “g,k S,X×S), which is an Artin stack locally of ﬁnite presentation over Sby Proposition 2.11 in [25] since S is an excellent scheme over C and X× S is the quotient stack [W×S/G]. Thus, by Lemma C.5 in [2] we conclude the proof. 2.4.2. Automorphisms. So far we have shown that Qε g,k(X,β) is an Artin s”
QUANTUM KIRWAN MORPHISM AND GROMOV-WITTEN INVARIANTS OF QUOTIENTS I (2012, methodology): “3 so that X = C2 ⊕ C3 and X//G = P[2, 3].”
Logarithmic Geometry and Moduli (2010, methodology): “Twisted curves are a central object in the theory of twisted stable maps [2, 5, 1]: in order to have a complete moduli space of stable maps C → X of type Γ, where X is a proper tame stack with projective coarse moduli space and Γ = (g, n, β) are the relevant discrete data, one must allow the curve C”
Stable rationality and conic bundles (2015, background): “Then there is a relative notion of corase moduli space, introduced in [4], which determines a stack with representable morphism to W , through which f factors.”
The étale local structure of algebraic stacks (2019, result): “tabilizers, is ´etale locally a quotient stack around any point with a linearly reductive stabilizer. This result generalizes the main result of [AHR19] to the relative setting and the main result of [AOV11] to the case of non-ﬁnite inertia. We also provide various coherent completeness and eﬀectivi”
Stable quasimaps to GIT quotients (2011, background): “While the fact that Qmapg,k(W/G, β) is an Artin stack locally of finite type follows quite easily by combining some results in [32] and [1], we give a straightforward direct construction which exhibits it as a stack with a schematic morphism toBunG, whose fibers”