Valuations and asymptotic invariants for sequences of ideals
By Mattias Jonsson, M. Mustaţă. 2010
253 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/488883f99939640b82a8466db89e9d29f8acd122
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In Algebra and Number Theory
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Citations in the corpus
- Singular semipositive metrics in non-Archimedean geometry (2011, background): “ally principal and D2Div0(X0) is the corresponding eective Cartier divisor. SINGULAR SEMIPOSITIVE METRICS IN NON-ARCHIMEDEAN GEOMETRY 45 This denition only depends on the model function clogjaj(cf. [JM12]), and would in fact make sense for an arbitrary non-positive model function ’2D(X). If ais a gr”
- K-semistability is equivariant volume minimization (2015, methodology): “oordinates fz 1;:::;z rgsuch that z idenes at an irreducible component D iof D. We put QM(Y;D) = [ QM (Y;D). From now on in this paper, we assume that Xhas Q-Gorenstein klt singularity. Following [JM10] and [BFFU13], we can dene the log discrepancy for any valuation v2Val X. This is achieved in thr”
- Effectiveness of Demailly’s strong openness conjecture and related problems (2014, background): “In [21], we solved the following conjecture about the volumes growth of the sublevel sets of quasi-plurisubharmonic functions posed by Jonsson and Mustată in [26] (see also [25]): Conjecture J-M: Let ψ be a plurisubharmonic function on n ⊂ Cn .”
- A proof of Demailly's strong openness conjecture (2015, background): “For dimX ≤ 2, the strong openness conjecture was proved in [16] by studying the asymptotic jumping numbers for the graded sequences of ideals.”
- An algebraic approach to the openness conjecture of Demailly and Kollár (2012, methodology): “It was shown in [19] that this more general conjecture follows from the special case in Conjecture C above.”
- On the Yau‐Tian‐Donaldson Conjecture for Singular Fano Varieties (2017, unknown): “By [29], w → w(l(mΦ)) and w → w(B × C) are continuous, while w 7→ A(M,B)×C(w) is lower semicontinuous on W .”
- Thresholds, valuations, and K-stability (2017, methodology): “his theorem can be viewed as a global analogue of the main result in [Blu18a], where the existence of a valuation minimizing the normalized volume is established. It is also reminiscent of results in [JM12] on the existence of valuations computing log canonical thresholds of graded sequence of ideal”
- A Guided Tour to Normalized Volume (2018, background): “orial. Proof. We first consider the case when X = C2. Let v∗be a minimizer and denote a•= {am(v∗)}m∈N. Then it was known that v∗computes the log canonical threshold of (X,D+a•). By similar argument as [JM12], we know that v∗must be quasi-monomial. If v∗is divisorial, then we know that the associated ”
- A minimizing valuation is quasi-monomial (2019, unknown): “n r. Let (y1,···,yr) be a system of parameters of OY,η at ηsuch that Ei j = (yj = 0). Then for any α= (α1,···,αr) ∈Rr ≥0 \{0}, we define A(X,∆)(vα) as AX,∆(vα) := Xr j=1 αjAX,∆(ordE ij ). (2.2) (3) In [JM12], it was showed that there exists a retraction map ρY : ValX →QM(Y,E) for any log smooth model”