scholarly journals Lowest log canonical thresholds of a reduced plane curve of degree d

2019 ◽  
Vol 6 (4) ◽  
pp. 1216-1235
Author(s):  
Nivedita Viswanathan
Keyword(s):  
Plane Curve ◽  
Log Canonical Threshold ◽  
Log Canonical

AbstractWe describe the sixth worst singularity that a plane curve of degree $$d\geqslant 5$$ d ⩾ 5 could have, using its log canonical threshold at the point of singularity. This is an extension of a result due to Cheltsov (J Geom Anal 27(3):2302–2338, 2017) wherein the five lowest values of log canonical thresholds of a plane curve of degree $$d \geqslant 3$$ d ⩾ 3 were computed. These six small log canonical thresholds, in order, are 2 / d, $$({2d-3})/{(d-1)^2}$$ ( 2 d - 3 ) / ( d - 1 ) 2 , $$({2d-1})/(d^2-d)$$ ( 2 d - 1 ) / ( d 2 - d ) , $$({2d-5})/({d^2-3d+1})$$ ( 2 d - 5 ) / ( d 2 - 3 d + 1 ) , $$({2d-3})/(d^2-2d)$$ ( 2 d - 3 ) / ( d 2 - 2 d ) and $$({2d-7})/({d^2-4d+1})$$ ( 2 d - 7 ) / ( d 2 - 4 d + 1 ) . We give examples of curves with these values as their log canonical thresholds using illustrations.

scholarly journals The log-canonical threshold of a plane curve

2016 ◽  
Vol 160 (3) ◽  
pp. 513-535 ◽  
Author(s):  
CARLOS GALINDO ◽  
FERNANDO HERNANDO ◽  
FRANCISCO MONSERRAT
Keyword(s):  
Explicit Formula ◽  
Plane Curve ◽  
Log Canonical Threshold ◽  
Log Canonical ◽  
Maximal Contact ◽  
Intersection Multiplicities

AbstractWe give an explicit formula for the log-canonical threshold of a reduced germ of plane curve. The formula depends only on the first two maximal contact values of the branches and their intersection multiplicities. We also improve the two branches formula given in [27].

scholarly journals JET SCHEMES OF QUASI-ORDINARY SURFACE SINGULARITIES

2019 ◽  
pp. 1-88
Author(s):  
HELENA COBO ◽  
HUSSEIN MOURTADA
Keyword(s):  
Plane Curve ◽  
Topological Type ◽  
Weighted Graph ◽  
Singular Locus ◽  
Hypersurface Singularity ◽  
Log Canonical Threshold ◽  
Jet Schemes ◽  
Log Canonical ◽  
Irreducible Components ◽  
Divisorial Valuation

We describe the irreducible components of the jet schemes with origin in the singular locus of a two-dimensional quasi-ordinary hypersurface singularity. A weighted graph is associated with these components and with their embedding dimensions and their codimensions in the jet schemes of the ambient space. We prove that the data of this weighted graph is equivalent to the data of the topological type of the singularity. We also determine a component of the jet schemes (equivalent to a divisorial valuation on $\mathbb{A}^{3}$ ), that computes the log-canonical threshold of the singularity embedded in $\mathbb{A}^{3}$ . This provides us with pairs $X\subset \mathbb{A}^{3}$ whose log-canonical thresholds are not computed by monomial divisorial valuations. Note that for a pair $C\subset \mathbb{A}^{2}$ , where $C$ is a plane curve, the log-canonical threshold is always computed by a monomial divisorial valuation (in suitable coordinates of $\mathbb{A}^{2}$ ).

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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