Arc valuations on smooth varieties
by More, Yogesh K., Ph.D., UNIVERSITY OF MICHIGAN, 2008, 80 pages; 3328915

Abstract:

For a nonsingular k-arc valuation v on a nonsingular variety X over a field k, we describe the maximal irreducible subset C( v) of the arc space of X such that valC(v) = v. We describe C(v) both algebraically, in terms of the sequence of valuation ideals of v, and geometrically, in terms of the sequence of infinitely near points associated to v. For a singular k-arc valuation v, we show that after a finite number of blowups of centers, its becomes nonsingular. When X is a surface, our construction also applies to any divisorial valuation v, and in this case C(v) coincides with the construction of Ein, Lazarsfeld, and Mustata ([7, Example 2.5]). We also investigate the situation for irrational valuations on surfaces. Our results suggest that a more natural place to look for these valuations are in spaces that generalize arc spaces. Also, we compute the motivic measure of C(v) for some of the various types of valuations on surfaces.

 
AdviserKaren E. Smith
SchoolUNIVERSITY OF MICHIGAN
SourceDAI/B 69-09, p. , Dec 2008
Source TypeDissertation
SubjectsMathematics
Publication Number3328915
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