Valuations and asymptotic invariants for sequences of ideals

@article{Jonsson2010ValuationsAA,
  title={Valuations and asymptotic invariants for sequences of ideals},
  author={Mattias Jonsson and Mircea Mustaţǎ},
  journal={arXiv: Algebraic Geometry},
  year={2010}
}
We study asymptotic jumping numbers for graded sequences of ideals, and show that every such invariant is computed by a suitable real valuation of the function field. We conjecture that every valuation that computes an asymptotic jumping number is necessarily quasi-monomial. This conjecture holds in dimension two. In general, we reduce it to the case of affine space and to graded sequences of valuation ideals. Along the way, we study the structure of a suitable valuation space. 
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References

SHOWING 1-10 OF 78 REFERENCES
Jumping coefficients of multiplier ideals
We study in this paper some local invariants attached via multiplier ideals to an effective divisor or ideal sheaf on a smooth complex variety. First considered (at least implicitly) by Libgober and
Log canonical thresholds on varieties with bounded singularities
We consider pairs (X,A), where X is a variety with klt singularities and A is a formal product of ideals on X with exponents in a fixed set that satisfies the Descending Chain Condition. We also
Valuations and multiplier ideals
This article is the third of a series of work on a new approach to the study of singularities of various objects in a local, two-dimensional setting. Our focus in the present paper is on multiplier
Invariants of singularities of pairs
Let X be a smooth complex variety and Y be a closed subvariety of X, or more generally, a closed subscheme of X. We are interested in invariants attached to the singularities of the pair (X, Y ). We
Singular semipositive metrics in non-Archimedean geometry
Let X be a smooth projective Berkovich space over a complete discrete valuation field K of residue characteristic zero, endowed with an ample line bundle L. We introduce a general notion of (possibly
Good formal structures for flat meromorphic connections, II: Excellent schemes
Given a flat meromorphic connection on an excellent scheme over a field of characteristic zero, we prove existence of good formal structures after blowing up; this extends a theorem of Mochizuki for
Valuations and plurisubharmonic singularities
We extend to higher dimensions some of the valuative analysis of singularities of plurisubharmonic (psh) functions developed by the first two authors. Following Kontsevich and Soibelman we describe
Multiplier ideals and integral closure of monomial ideals: An analytic approach
Proofs of two results about a monomial ideal -- describing membership in auxiliary ideals associated to the monomial ideal -- are given which do not invoke resolution of singularities. The AM--GM
...
1
2
3
4
5
...