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182
Algebraic orbifold quantum products
"... The purpose of this note is to give an overview of our work on defining algebraic counterparts for W. Chen and Y. Ruan’s GromovWitten Theory of orbifolds. This work will be described in detail in a subsequent paper. The presentation here is generally based on lectures given by two of us at the Orbi ..."
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Cited by 41 (1 self)
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The purpose of this note is to give an overview of our work on defining algebraic counterparts for W. Chen and Y. Ruan’s GromovWitten Theory of orbifolds. This work will be described in detail in a subsequent paper. The presentation here is generally based on lectures given by two of us at the Orbifold Workshop
Moduli of Twisted Spin Curves
"... . In this note we give a new, natural construction of a compactication of the stack of smooth rspin curves, which we call the stack of stable twisted rspin curves. This stack is identied with a special case of a stack of twisted stable maps of Abramovich and Vistoli. Realizations in terms of ad ..."
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Cited by 39 (7 self)
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. In this note we give a new, natural construction of a compactication of the stack of smooth rspin curves, which we call the stack of stable twisted rspin curves. This stack is identied with a special case of a stack of twisted stable maps of Abramovich and Vistoli. Realizations in terms of admissible G m spaces and Qline bundles are given as well. The innitesimal structure of this stack is described in a relatively straightforward manner, similar to that of usual stable curves. We construct representable morphisms from the stacks of stable twisted rspin curves to the stacks of stable rspin curves [4], and show that they are isomorphisms. Many delicate features of rspin curves, including torsion free sheaves with power maps, arise as simple byproducts of twisted spin curves. Various constructions, such as the @operator of Seeley and Singer [9] and Witten's cohomology class [10] go through without complications in the setting of twisted spin curves. The moduli s...
KawamataViehweg vanishing as Kodaira vanishing for stacks
"... We associate to a pair (X, D), consisting of a smooth variety with a divisor D ∈ Div(X) ⊗ Q whose support has only 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. ..."
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Cited by 35 (3 self)
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We associate to a pair (X, D), consisting of a smooth variety with a divisor D ∈ Div(X) ⊗ Q whose support has only 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.
Twisted jet, motivic measure and orbifold cohomology
, 2001
"... We introduce the notion of twisted jet. For a DeligneMumford stack X of finite type over an algebraically closed field k, a twisted ∞jet on X is a representable morphism D → X such that D is a smooth DeligneMumford stack with the coarse moduli space Spec k[[t]]. We study the motivic measure on t ..."
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Cited by 34 (2 self)
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We introduce the notion of twisted jet. For a DeligneMumford stack X of finite type over an algebraically closed field k, a twisted ∞jet on X is a representable morphism D → X such that D is a smooth DeligneMumford stack with the coarse moduli space Spec k[[t]]. We study the motivic measure on the space of the twisted ∞jets on a smooth DeligneMumford stack. As an application, we prove that two birational minimal models with Gorenstein quotient singularities have the same orbifold cohomology with Hodge structure. Our main results are Theorem 0.4 and 0.5.
Log canonical models for the moduli space of curves: First divisorial contraction
, 2007
"... In this paper, we initiate our investigation of log canonical models for (Mg, αδ) as we decrease α from 1 to 0. We prove that for the first critical value α = 9/11, the log canonical model is isomorphic to the moduli space of pseudostable curves, which have nodes and cusps as singularities. We als ..."
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Cited by 33 (3 self)
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In this paper, we initiate our investigation of log canonical models for (Mg, αδ) as we decrease α from 1 to 0. We prove that for the first critical value α = 9/11, the log canonical model is isomorphic to the moduli space of pseudostable curves, which have nodes and cusps as singularities. We also show that α = 7/10 is the next critical value, i.e., the log canonical model stays the same in the interval (7/10, 9/11]. In the appendix, we develop a theory of log canonical models of stacks that explains how these can be expressed in terms of the coarse moduli space.
The orbifold quantum cohomology of the classifying space of a finite group
 Orbifolds in Mathematics and Physics, Contemp
, 2002
"... Abstract. We work through, in detail, the quantum cohomology of the orbifold BG, the point with action of a finite group G. We provide a simple description of algebraic structures on the state space of this theory. As a consequence, we find that multiple copies of commuting Virasoro algebras appear ..."
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Cited by 31 (2 self)
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Abstract. We work through, in detail, the quantum cohomology of the orbifold BG, the point with action of a finite group G. We provide a simple description of algebraic structures on the state space of this theory. As a consequence, we find that multiple copies of commuting Virasoro algebras appear in this theory which completely determine the correlators of the theory. For the Proceedings of the Mathematical Aspects of Orbifold String Theory conference in Madison, Wisconsin.
ORBIFOLD QUANTUM RIEMANNROCH, LEFSCHETZ AND SERRE
, 2009
"... Given a vector bundle F on a smooth DeligneMumford stack X and an invertible multiplicative characteristic class c, we define orbifold GromovWitten invariants of X twisted by F and c. We prove a “quantum RiemannRoch theorem” (Theorem 4.2.1) which expresses the generating function of the twisted i ..."
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Cited by 30 (9 self)
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Given a vector bundle F on a smooth DeligneMumford stack X and an invertible multiplicative characteristic class c, we define orbifold GromovWitten invariants of X twisted by F and c. We prove a “quantum RiemannRoch theorem” (Theorem 4.2.1) which expresses the generating function of the twisted invariants in terms of the generating function of the untwisted invariants. A quantum Lefschetz hyperplane theorem is derived from this by specializing to genus zero. As an application, we determine the relationship between genus0 orbifold GromovWitten invariants of X and that of a complete intersection, under additional assumptions. This provides a way to verify mirror symmetry predictions for some complete intersection orbifolds.
Homstacks and restriction of scalars
 Duke Math. J
"... 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 finite ..."
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Cited by 29 (3 self)
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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. 1. Statements of results Fix an algebraic space S, let X and Y be separated Artin stacks of finite presentation over
On covering of DeligneMumford stacks and surjectivity of the Brauer map
 Bull. London Math. Soc
"... This paper is concerned with finite covers of Deligne–Mumford stacks by schemes, in connection with the theory of Brauer group. The reader is referred to [6] for basic references on algebraic stacks and Brauer groups. We are primarily concerned with Deligne–Mumford stacks; every ..."
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Cited by 28 (1 self)
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This paper is concerned with finite covers of Deligne–Mumford stacks by schemes, in connection with the theory of Brauer group. The reader is referred to [6] for basic references on algebraic stacks and Brauer groups. We are primarily concerned with Deligne–Mumford stacks; every
ON (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 u ..."
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Cited by 27 (0 self)
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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.