scholarly journals F-regular and F-pure rings vs. log terminal and log canonical singularities

2002 ◽  
Vol 11 (2) ◽  
pp. 363-392 ◽  
Author(s):  
Nobuo Hara ◽  
Kei-ichi Watanabe
Keyword(s):  
Log Canonical ◽  
Canonical Singularities

scholarly journals On minimal log discrepancies and kollár components

2021 ◽  
pp. 1-20
Author(s):  
Joaquín Moraga
Keyword(s):  
Blow Up ◽  
Chain Condition ◽  
Fano Varieties ◽  
Finite Sets ◽  
Fixed Dimension ◽  
Log Canonical ◽  
Finite Set ◽  
Canonical Singularities ◽  
Ascending Chain Condition

Abstract In this article, we prove a local implication of boundedness of Fano varieties. More precisely, we prove that $d$ -dimensional $a$ -log canonical singularities with standard coefficients, which admit an $\epsilon$ -plt blow-up, have minimal log discrepancies belonging to a finite set which only depends on $d,\,a$ and $\epsilon$ . This result gives a natural geometric stratification of the possible mld's in a fixed dimension by finite sets. As an application, we prove the ascending chain condition for minimal log discrepancies of exceptional singularities. We also introduce an invariant for klt singularities related to the total discrepancy of Kollár components.

scholarly journals On the -purity of isolated log canonical singularities

2013 ◽  
Vol 149 (9) ◽  
pp. 1495-1510 ◽  
Author(s):  
Osamu Fujino ◽  
Shunsuke Takagi
Keyword(s):  
Characteristic Zero ◽  
Three Dimensional ◽  
Canonical Singularity ◽  
Pure Type ◽  
Log Canonical ◽  
Canonical Singularities ◽  
Log Canonical Singularity

AbstractA singularity in characteristic zero is said to be of dense $F$-pure type if its modulo $p$ reduction is locally Frobenius split for infinitely many $p$. We prove that if $x\in X$ is an isolated log canonical singularity with $\mu (x\in X)\leq 2$ (where the invariant $\mu $ is as defined in Definition 1.4), then it is of dense $F$-pure type. As a corollary, we prove the equivalence of log canonicity and being of dense $F$-pure type in the case of three-dimensional isolated $ \mathbb{Q} $-Gorenstein normal singularities.

KAWACHI'S INVARIANT FOR NORMAL SURFACE SINGULARITIES

1998 ◽  
Vol 09 (05) ◽  
pp. 623-640 ◽  
Author(s):  
VLADIMIR MAŞEK
Keyword(s):  
Linear Systems ◽  
A Priori ◽  
Normal Surface ◽  
Numerical Invariant ◽  
Normal Surfaces ◽  
Log Canonical ◽  
Surface Singularities ◽  
Canonical Surface ◽  
Quick Proof ◽  
Canonical Singularities

We study a useful numerical invariant of normal surface singularities, introduced recently by T. Kawachi. Using this invariant, we give a quick proof of the (well-known) fact that all log-canonical surface singularities are either elliptic Gorenstein or rational (without assuming a priori that they are ℚ-Gorenstein). In Sec. 2 we prove effective results (stated in terms of Kawachi's invariant) regarding global generation of adjoint linear systems on normal surfaces with boundary. Such results can be used in proving effective estimates for global generation on singular threefolds. The theorem of Ein–Lazarsfeld and Kawamata, which says that the minimal center of log-canonical singularities is always normal, explains why the results proved here are relevant in that situation.

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