Let X be an algebraic stack in the sense of Deligne-Mumford. We construct a purely 0-dimensional algebraic stack over X (in the sense of Artin), the intrinsic normal cone C-x. The notion of (perfect) obstruction theory for X is introduced, and it is shown how to construct, given a perfect obstruction theory for X, a pure-dimensional virtual fundamental class in the Chow group of X. We then prove some properties of such classes, both in the absolute and in the relative context, Via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory, and sometimes a perfect one.

The intrinsic normal cone / Behrend, K.; Fantechi, Barbara. - In: INVENTIONES MATHEMATICAE. - ISSN 0020-9910. - 128:1(1997), pp. 45-88. [10.1007/s002220050136]

The intrinsic normal cone

Fantechi, Barbara
1997-01-01

Abstract

Let X be an algebraic stack in the sense of Deligne-Mumford. We construct a purely 0-dimensional algebraic stack over X (in the sense of Artin), the intrinsic normal cone C-x. The notion of (perfect) obstruction theory for X is introduced, and it is shown how to construct, given a perfect obstruction theory for X, a pure-dimensional virtual fundamental class in the Chow group of X. We then prove some properties of such classes, both in the absolute and in the relative context, Via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory, and sometimes a perfect one.
1997
128
1
45
88
Behrend, K.; Fantechi, Barbara
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/16663
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