On weak Fano varieties with log canonical singularities

TL;DR

This paper investigates the semi-ampleness of anti-canonical divisors in weak Fano varieties with log canonical singularities, providing new conditions for semi-ampleness and exploring their geometric properties.

Contribution

It establishes semi-ampleness results for weak Fano 3-folds and 4-folds with log canonical singularities, introduces counterexamples, and proposes optimal conditions for semi-ampleness.

Findings

Semi-ampleness holds for weak Fano 3-folds with log canonical singularities.

Counterexamples show semi-ampleness does not always hold.

Sufficient conditions for semi-ampleness are proposed and proven.

Abstract

We prove that the anti-canonical divisors of weak Fano 3-folds with log canonical singularities are semiample. Moreover, we consider semiampleness of the anti-log canonical divisor of any weak log Fano pair with log canonical singularities. We show semiampleness dose not hold in general by constructing several examples. Based on those examples, we propose sufficient conditions which seem to be the best possible and we prove semiampleness under such conditions. In particular we derive semiampleness of the anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers are at most 1-dimensional. We also investigate the Kleiman-Mori cones of weak log Fano pairs with log canonical singularities.