On pluri-half-anticanonical systems of LeBrun twistor spaces

Abstract:

In this paper, we investigate pluri-half-anticanonical systems on the so-called LeBrun twistor spaces. We determine its dimension, the base locus, the structure of the associated rational map, and also the structure of general members, in precise form. In particular, we show that if $n\ge 3$ and $m\ge 2$, the base locus of the system $|mK^{-1/2}|$ on $n\mathbb {CP}^2$ 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 member of $|mK^{-1/2}|$ becomes an irreducible non-singular surface. We also show that if $n\ge 4$ and $m\ge n-1$, 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 $m\ge n-1$.