Q-complements on log surfaces

TL;DR

This paper classifies log surfaces without $ ext{Q}$-complements, showing they are always non-rational, thereby broadening the applicability of complement theory to a wider class of log surfaces.

Contribution

It provides a complete classification of log surfaces lacking $ ext{Q}$-complements, removing previous restrictions and expanding the scope of complement theory.

Findings

Log surfaces without $ ext{Q}$-complements are always non-rational.

The classification broadens the applicability of complement theory.

The results remove previous restrictions in the theory of complements.

Abstract

In this paper the log surfaces without $\QQ$-complement are classified. In particular, they are non-rational always. This result takes off the restriction in the theory of complements and allows one to apply it in the most wide class of log surfaces.