A Transversality Theorem for some Classical Varieties

TL;DR

This paper extends Kleiman's transversality theorem to varieties with specific singularities and demonstrates that several classical varieties possess log terminal singularities, broadening the understanding of their geometric properties.

Contribution

It generalizes a fundamental transversality theorem to a wider class of singular varieties and classifies certain classical varieties as log terminal.

Findings

Kleiman's transversality theorem is extended to log terminal and log canonical varieties.

Certain classical varieties are proven to have log terminal singularities.

The work broadens the scope of transversality and singularity classification in algebraic geometry.

Abstract

In 2009, de Fernex and Hacon proposed a generalization of the notion of the singularities to normal varieties that are not Q-Gorenstein. Based on their work, we generalize Kleiman's transversality theorem to subvarieties with log terminal or log canonical singularities. We also show that some classical varieties, such as generic determinantal varieties, W^r_d for general smooth curves, and certain Schubert varieties in G(k, n) are log terminal.