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43
Germs of Arcs on Singular Algebraic Varieties and Motivic Integration
, 1999
"... Introduction Let k be a field of characteristic zero. We denote by M the Grothendieck ring of algebraic varieties over k (i.e. reduced separated schemes of finite type over k). It is the ring generated by symbols [S], for S an algebraic variety over k, with the relations [S] = [S 0 ] if S is iso ..."
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Cited by 151 (20 self)
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Introduction Let k be a field of characteristic zero. We denote by M the Grothendieck ring of algebraic varieties over k (i.e. reduced separated schemes of finite type over k). It is the ring generated by symbols [S], for S an algebraic variety over k, with the relations [S] = [S 0 ] if S is isomorphic to S 0 , [S] = [S n S 0 ] + [S 0 ] if S 0 is closed in S and [S \Theta S 0 ] = [S] [S 0 ]. Note that, for S an algebraic variety over k, the mapping S 0 7! [S 0 ] from the
Inversion of adjunction for local complete intersection varieties
 AMER. J. MATH
, 2003
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Motivic integration and the grothendieck group of pseudofinite fields
 Proceedings of the International Congress of Mathematicians (ICM 2002
"... Motivic integration is a powerful technique to prove that certain quantities associated to algebraic varieties are birational invariants or are independent of a chosen resolution of singularities. We survey our recent work on an extension of the theory of motivic integration, called arithmetic motiv ..."
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Cited by 18 (4 self)
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Motivic integration is a powerful technique to prove that certain quantities associated to algebraic varieties are birational invariants or are independent of a chosen resolution of singularities. We survey our recent work on an extension of the theory of motivic integration, called arithmetic motivic integration. We developed this theory to understand how padic integrals of a very general type depend on p. Quantifier elimination plays a key role.
Motivic integration on smooth rigid varieties and invariants of degenerations
 Duke Math. J
"... We develop a theory of motivic integration for smooth rigid varieties. As an application we obtain a motivic analogue for rigid varieties of Serre’s invariant for padic varieties. Our construction provides new geometric birational invariants of degenerations of algebraic varieties. For degeneration ..."
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Cited by 16 (2 self)
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We develop a theory of motivic integration for smooth rigid varieties. As an application we obtain a motivic analogue for rigid varieties of Serre’s invariant for padic varieties. Our construction provides new geometric birational invariants of degenerations of algebraic varieties. For degenerations of CalabiYau varieties, our results take a stronger form. 1.
Applications of patching to quadratic forms and central simple algebras, preprint arXiv:0809.4481
"... Abstract. This paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields. In particular, we use a patching approach to reprove and generalize a recent result of Parimala and Suresh on the uinvariant of padic ..."
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Cited by 13 (3 self)
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Abstract. This paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields. In particular, we use a patching approach to reprove and generalize a recent result of Parimala and Suresh on the uinvariant of padic function fields, p 6 = 2. The strategy relies on a localglobal principle for homogeneous spaces for rational algebraic groups, combined with local computations. 1.
Motivic integration over DeligneMumford stacks
, 2004
"... The aim of this article is to develop the theory of motivic integration over DeligneMumford stacks and to apply it to the birational geometry of DeligneMumford stacks. ..."
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Cited by 11 (0 self)
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The aim of this article is to develop the theory of motivic integration over DeligneMumford stacks and to apply it to the birational geometry of DeligneMumford stacks.
Jet schemes and singularities
"... The study of singularities of pairs is fundamental for higher dimensional birational geometry. The usual approach to invariants of such singularities is via divisorial valuations, as in [Kol]. In this paper we give a selfcontained presentation of an alternative approach, via contact loci in spaces ..."
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Cited by 11 (2 self)
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The study of singularities of pairs is fundamental for higher dimensional birational geometry. The usual approach to invariants of such singularities is via divisorial valuations, as in [Kol]. In this paper we give a selfcontained presentation of an alternative approach, via contact loci in spaces of arcs. Our main application is