Finite determination conjecture for Mather-Jacobian minimal log discrepancies and its applications

Shihoko Ishii

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Finite determination conjecture for Mather-Jacobian minimal log discrepancies and its applications

Shihoko Ishii

Algebraic Geometry · Submitted 24 Mar 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We also show that this conjecture yields some basic properties of singularities; e.g. openness of Mather-Jacobian (log) canonical singularities, stability of these singularities under small deformations and lower semi-continuity of Mather-Jacobian minimal log discrepancies, which are already known in characteristic 0 and open for positive characteristic case.We show some evidences of the conjecture: for example, for non-degenerate hypersurfaces of any dimension in arbitrary characteristic and 2-dimensional singularities in characteristic not 2.

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CitationShihoko Ishii. "Finite determination conjecture for Mather-Jacobian minimal log discrepancies and its applications." arXiv:1703.08500 (2017). https://pith.science/paper/IIFUJFF5

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.54 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.70 · compares with prior results 0.78 · mentions code or data availability 0.02.

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In this paper we study singularities in arbitrary characteristic. We propose Finite Determination Conjecture for Mather-Jacobian minimal log discrepancies in terms of jet schemes of a singularity. The conjecture is equivalent to the boundedness of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy. We also show that this conjecture yields some basic properties of singularities; e.g. openness of Mather-Jacobian (log) canonical singularities, stability of these singularities under small deformations and lower semi-continuity of Mather-Jacobian minimal log discrepancies, which are already known in characteristic 0 and open for positive characteristic case.We show some evidences of the conjecture: for example, for non-degenerate hypersurfaces of any dimension in arbitrary characteristic and 2-dimensional singularities in characteristic not 2. We aslo give a bound of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this paper we study singularities in arbitrary characteristic.	background 0.67	0.60	0.02	asserted 0.85
s1	We propose Finite Determination Conjecture for Mather-Jacobian minimal log discrepancies in terms of jet schemes of a singularity.	contribution 1.00	0.93	0.17	speculative 0.70
s2	The conjecture is equivalent to the boundedness of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy.	result 0.55	0.58	0.19	asserted 1.00
s3	We also show that this conjecture yields some basic properties of singularities; e.g. openness of Mather-Jacobian (log) canonical singularities, stability of these singularities under small deformations and lower semi-continuity of Mather-Jacobian minimal log discrepancies, which are already known in characteristic 0 and open for positive characteristic case.We show some evidences of the conjecture: for example, for non-degenerate hypersurfaces of any dimension in arbitrary characteristic and 2-dimensional singularities in characteristic not 2.	result 0.56	0.89	0.33	asserted 0.99
s4	We aslo give a bound of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy.	result 0.82	0.94	0.18	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.83 · evidence from proof or derivation 0.98.

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Original abstract

In this paper we study singularities in arbitrary characteristic. We propose Finite Determination Conjecture for Mather-Jacobian minimal log discrepancies in terms of jet schemes of a singularity. The conjecture is equivalent to the boundedness of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy. We also show that this conjecture yields some basic properties of singularities; e.g. openness of Mather-Jacobian (log) canonical singularities, stability of these singularities under small deformations and lower semi-continuity of Mather-Jacobian minimal log discrepancies, which are already known in characteristic 0 and open for positive characteristic case.We show some evidences of the conjecture: for example, for non-degenerate hypersurfaces of any dimension in arbitrary characteristic and 2-dimensional singularities in characteristic not 2. We aslo give a bound of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy.

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* The proof of [ref] can be interpreted into the discussion of Newton polygon and we can prove the above theorem in the same way as [ref] using Maclagan's result.

A boundedness conjecture for minimal log discrepancies on a fixed germ · Some affirmative cases · 0.94

Cites as background 2 papers

Mather-Jacobian log discrepancy is defined in [ref] and [ref] independently, for a variety over a field of characteristic zero and is easily generalized to positive characteristic case (see, for example, [ref] ).

Singularities in arbitrary characteristic via jet schemes · Preliminaries on Mather-Jacobian log discrepancies and jet schemes · 0.93
Note that this conjecture is not yet proved in general even for characteristic zero (this is viewed as a special case of Conjecture 1.1 in [ref] ).

A boundedness conjecture for minimal log discrepancies on a fixed germ · Introduction · 0.81

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