scholarly journals HODGE IDEALS FOR -DIVISORS, -FILTRATION, AND MINIMAL EXPONENT

2020 ◽  
Vol 8 ◽  
Author(s):  
MIRCEA MUSTAŢĂ ◽  
MIHNEA POPA
Keyword(s):  
Log Canonical Threshold ◽  
Basic Properties ◽  
General Properties ◽  
Log Canonical ◽  
Refined Version

We compute the Hodge ideals of $\mathbb{Q}$ -divisors in terms of the $V$ -filtration induced by a local defining equation, inspired by a result of Saito in the reduced case. We deduce basic properties of Hodge ideals in this generality, and relate them to Bernstein–Sato polynomials. As a consequence of our study we establish general properties of the minimal exponent, a refined version of the log canonical threshold, and bound it in terms of discrepancies on log resolutions, addressing a question of Lichtin and Kollár.

scholarly journals Learning Coefficient of Vandermonde Matrix-Type Singularities in Model Selection

Entropy ◽  
2019 ◽  
Vol 21 (6) ◽  
pp. 561
Author(s):  
Miki Aoyagi
Keyword(s):  
Model Selection ◽  
Information Criteria ◽  
Learning Systems ◽  
Vandermonde Matrix ◽  
Selection Methods ◽  
Learning Models ◽  
Matrix Type ◽  
Log Canonical Threshold ◽  
Log Canonical ◽  
Blowing Up

In recent years, selecting appropriate learning models has become more important with the increased need to analyze learning systems, and many model selection methods have been developed. The learning coefficient in Bayesian estimation, which serves to measure the learning efficiency in singular learning models, has an important role in several information criteria. The learning coefficient in regular models is known as the dimension of the parameter space over two, while that in singular models is smaller and varies in learning models. The learning coefficient is known mathematically as the log canonical threshold. In this paper, we provide a new rational blowing-up method for obtaining these coefficients. In the application to Vandermonde matrix-type singularities, we show the efficiency of such methods.

scholarly journals Canonical and log canonical thresholds of multiple projective spaces

2019 ◽  
Author(s):  
Aleksandr V. Pukhlikov
Keyword(s):  
Projective Space ◽  
Projective Spaces ◽  
Regularity Conditions ◽  
Large Classes ◽  
Log Canonical Threshold ◽  
Birational Rigidity ◽  
Log Canonical ◽  
Finite Set ◽  
Dimensional Projective Space ◽  
Almost All

AbstractWe show that the global (log) canonical threshold of d-sheeted covers of the M-dimensional projective space of index 1, where $$d\geqslant 4$$d⩾4, is equal to 1 for almost all families (except for a finite set). The varieties are assumed to have at most quadratic singularities, the rank of which is bounded from below, and to satisfy the regularity conditions. This implies birational rigidity of new large classes of Fano–Mori fibre spaces over a base, the dimension of which is bounded from above by a constant that depends (quadratically) on the dimension of the fibre only.

scholarly journals Multiplicities and log canonical threshold

2004 ◽  
Vol 13 (3) ◽  
pp. 603-615 ◽  
Author(s):  
Tommaso de Fernex ◽  
Lawrence Ein ◽  
Mircea Mustaţǎ
Keyword(s):  
Log Canonical Threshold ◽  
Log Canonical

scholarly journals A sharp lower bound for the log canonical threshold

2014 ◽  
Vol 212 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Jean-Pierre Demailly ◽  
Hoàng Hiệp Phạm
Keyword(s):  
Lower Bound ◽  
Log Canonical Threshold ◽  
Log Canonical ◽  
Sharp Lower Bound

scholarly journals Extremal cases for the log canonical threshold

2015 ◽  
Vol 353 (1) ◽  
pp. 21-24 ◽  
Author(s):  
Alexander Rashkovskii
Keyword(s):  
Log Canonical Threshold ◽  
Log Canonical
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