## The Intrinsic Normal Cone (1997)

Venue: | Invent. Math |

Citations: | 230 - 7 self |

### BibTeX

@ARTICLE{Behrend97theintrinsic,

author = {K. Behrend and B. Fantechi},

title = {The Intrinsic Normal Cone},

journal = {Invent. Math},

year = {1997},

volume = {128},

pages = {45--88}

}

### Years of Citing Articles

### OpenURL

### Abstract

We suggest a construction of virtual fundamental classes of certain types of moduli spaces. Contents 0

### Citations

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375 |
The irreducibility of the space of curves of given genus
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(Show Context)
Citation Context ...assume all algebraic stacks (in particular all algebraic spaces and all schemes) to be quasi-separated and locally of finite type over k. A Deligne-Mumford stack is an algebraic stack in the sense of =-=[5]-=-, in other words an algebraic stack with unramified diagonal. For a Deligne-Mumford stack X we denote by X fl the big fppf-site and by X 'et the small 'etale site of X . The associated topoi of sheave... |

231 |
Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete
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Citation Context ...ort exact sequence 0 \Gamma! F 0 \Gamma! F \Gamma! E \Gamma! 0 of coherent sheaves on X , with E locally free, then 0 \Gamma! C(E) \Gamma! C(F 0 ) \Gamma! C(F) \Gamma! 0 is exact, and conversely (see =-=[6]-=-, Example 4.1.7). Lemma 1.3 Let C ! D be a smooth, surjective morphism of cones, and let E = C \Theta D;0 X; then the sequence 0 \Gamma! E \Gamma! C \Gamma! D \Gamma! 0 is exact. Proof. Write C = Spec... |

184 |
Complexe cotangent et déformations
- Illusie
(Show Context)
Citation Context ... \Gamma! [I=I 2 ! f \Omega M ] in D(OU 'et ), where we think of [I=I 2 ! f \Omega M ] as a complex concentrated in degrees \Gamma1 and 0. Moreover, OE induces an isomorphism on h \Gamma1 and h 0 (see =-=[9]-=-, Chapitre III, Corollaire 3.1.3). Hence by Proposition 2.6 we get an induced isomorphism of cone stacks OE : [N U=M =f TM ] \Gamma! i NX ; where TM is the tangent bundle of M and N U=M is the normal ... |

162 | Functors on Artin rings
- Schlessinger
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(Show Context)
Citation Context ...ith kernel J isomorphic to k. A semi-small extension is one with kernel isomorphic to a k-vector space as an A 0 -algebra. Let F : Art ! Sets be a pro-representable covariant functor (in the sense of =-=[13]-=-). An obstruction space for F is a set k-vector space T 2 and, for any semi-small extension A 0 ! A with kernel J , an exact sequence F (A 0 ) \Gamma! F (A) ob \Gamma! T 2\Omega J: This means that, fo... |

159 |
Intersection theory on algebraic stacks and on their moduli spaces
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Citation Context ...ian cone N X=Y = Spec Sym I=I 2 is also called the normal sheaf of X in Y . More generally, any local immersion of Deligne-Mumford stacks has a normal cone whose abelian hull is its normal sheaf (see =-=[14]-=-, definition 1.20). 7 Exact Sequences of Cones Definition 1.2 A sequence of cone morphisms 0 \Gamma! E i \Gamma! C \Gamma! D \Gamma! 0 is exact if E is a vector bundle and locally over X there is a mo... |

150 |
Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert
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(Show Context)
Citation Context ...mooth of relative dimensions2, then E ffl is a perfect obstruction theory. 46 Spaces of Morphisms Let C and V be projective k-schemes. Let X = Mor(C; V ) be the k-scheme of morphisms from C to V (see =-=[7]-=-). Let f : C \Theta X ! V be the universal morphism ands: C \Theta X ! X the projection. By the functorial properties of the cotangent complex we get a homomorphism f L ffl V \Gamma! L ffl C \ThetaX \... |

149 | Gromov–Witten invariants in algebraic geometry
- Behrend
- 1997
(Show Context)
Citation Context ...onal (technical) assumption that it admits a global resolution, we can define a virtual fundamental class of the expected dimension. An application of the results of this work is contained in a paper =-=[3]-=- by the first author. There Gromov-Witten invariants are constructed for any genus, any target variety and the axioms listed in [4] are verified. We now give a more detailed outline of the contents of... |

99 |
Stacks of stable maps and Gromov-Witten invariants
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(Show Context)
Citation Context .... An application of the results of this work is contained in a paper [3] by the first author. There Gromov-Witten invariants are constructed for any genus, any target variety and the axioms listed in =-=[4]-=- are verified. We now give a more detailed outline of the contents of the paper. In the first section we recall what we need about cones and we introduce the notion of cone stacks over a Deligne-Mumfo... |

13 |
Enumeration of rational curves via torus actions, Preprint hep-th 9405035
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(Show Context)
Citation Context ...ven by a homomorphism of vector bundles F \Gamma1 ! F 0 . Then CX gives rise to a cone C in F 1 = F \Gamma1sand we intersect C with the zero section of F 1 . Another approach, suggested by Kontsevich =-=[11]-=-, is via virtual structure sheaves (see Remark 5.4). The drawback of that approach is that it requires a Riemann-Roch theorem for Deligne-Mumford stacks, for which we do not know a reference. In the s... |

12 |
Unobstructed deformations
- Kawamata
- 1995
(Show Context)
Citation Context ...(A 0 ) \Gamma! F (A) ob \Gamma! T 2\Omega J: This means that, for alls2 F (A),sis in the image of F (A 0 ) if and only if ob() = 0. It is also required that ob is functorial in the obvious sense (see =-=[10]-=-). We say that v 2 T 2 obstructs a small extension A 0 ! A if ob() = v\Omega w for somes2 F (A) and some nonzero w 2 J . Let X be a Deligne-Mumford stack, p 2 X a fixed point with residue field k. Let... |

9 |
Versal deformations and algebraic stacks, Inventiones Mathematicae 27
- Artin
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(Show Context)
Citation Context ... second author is a member of GNSAGA of CNR. Notations and Conventions Unless otherwise mentioned, we work over a fixed ground field k. An algebraic stack is an algebraic stack over k in the sense of =-=[1]-=- or [12]. Unless mentioned otherwise, we assume all algebraic stacks (in particular all algebraic spaces and all schemes) to be quasi-separated and locally of finite type over k. A Deligne-Mumford sta... |

6 |
Champs algébriques Preprint Université Paris 11
- Laumon, Moret-Bailly
- 1992
(Show Context)
Citation Context ... author is a member of GNSAGA of CNR. Notations and Conventions Unless otherwise mentioned, we work over a fixed ground field k. An algebraic stack is an algebraic stack over k in the sense of [1] or =-=[12]-=-. Unless mentioned otherwise, we assume all algebraic stacks (in particular all algebraic spaces and all schemes) to be quasi-separated and locally of finite type over k. A Deligne-Mumford stack is an... |

1 |
Algebraic Geometry---A First Course. Graduate Texts
- Harris
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(Show Context)
Citation Context ...s for I=mI . This defines a map f : W ! A r such that U is the fiber over the origin. Then C p is the normal cone to the image of W in A r . The proof then follows the argument of Proposition 20.2 in =-=[8]-=-. 2 5 Obstruction Theories and Fundamental Classes Virtual Fundamental Classes As usual, let X be a Deligne-Mumford stack over k. Definition 5.1 We call an obstruction theory E ffl ! L ffl X perfect, ... |