Behrend, Fantechi — The Intrinsic Normal Cone (1997)

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 \({\frak C}_X\) . 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.