Factoriality of nodal three-dimensional varieties and connectedness of the locus of log canonical singularities

2006 ◽  
Vol 197 (3) ◽  
pp. 387-414 ◽  
Author(s):  
I A Cheltsov
Keyword(s):  
Three Dimensional ◽  
Log Canonical ◽  
Canonical Singularities

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.

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.

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.

scholarly journals GENERAL HYPERPLANE SECTIONS OF THREEFOLDS IN POSITIVE CHARACTERISTIC

2018 ◽  
Vol 19 (2) ◽  
pp. 647-661 ◽  
Author(s):  
Kenta Sato ◽  
Shunsuke Takagi
Keyword(s):  
Projective Variety ◽  
Positive Characteristic ◽  
Hyperplane Section ◽  
Three Dimensional ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Double Points ◽  
Canonical Singularities ◽  
Hyperplane Sections

In this paper, we study the singularities of a general hyperplane section $H$ of a three-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>0$. We prove that if $X$ has only canonical singularities, then $H$ has only rational double points. We also prove, under the assumption that $p>5$, that if $X$ has only klt singularities, then so does $H$.

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