On pluri-half-anticanonical system of LeBrun twistor spaces

TL;DR

This paper studies the properties of pluri-half-anticanonical systems on LeBrun twistor spaces, revealing their dimensions, base loci, and the structure of associated rational maps and surfaces, with implications for minimal surface classification.

Contribution

It provides a detailed analysis of the pluri-half-anticanonical systems on LeBrun twistor spaces, including base locus structure, rational map behavior, and conditions for minimal surface types.

Findings

Base locus consists of two rational curves with singularities in general.

Blowing up these curves yields a non-singular surface.

The associated rational map is birational if and only if m > n-2.

Abstract

In this note, we investigate pluri-half-anticanonical systems on the so called LeBrun twistor spaces. We determine its dimension, the base locus, structure of the associated rational map, and also structure of general members, in precise form. In particular, we show that if n>2 and m>1, the base locus of the system |mK^{-1/2}| on nCP^2 consists of two non-singular rational curves, along which any member has singularity, and that if we blow-up these curves, then the strict transform of a general members of |mK^{-1/2}| becomes an irreducible non-singular surface. We also show that if n>3 and m>n-2, then the last surface is a minimal surface of general type with vanishing irregularity. We also show that the rational map associated to the system |mK^{-1/2}| is birational if and only if m> n-2.