Fixed Loci of the Anticanonical Complete Linear Systems of Anticanonical Rational Surfaces

TL;DR

This paper characterizes the fixed loci of the anticanonical linear system on anticanonical rational surfaces, providing explicit descriptions for geometrically ruled surfaces and confirming expected properties for smooth rational surfaces with positive self-intersection.

Contribution

It explicitly determines the fixed locus structure of the anticanonical linear system on certain rational surfaces, including the monoid of divisor classes and fixed prime divisors.

Findings

Fixed locus of the anticanonical linear system is characterized.

The monoid of nef divisor classes is explicitly generated by two elements.

Fixed prime divisors are $(-n)$-curves for some $n \\geq 1$.

Abstract

We determine the fixed locus of the anticanonical complete linear system of a given anticanonical rational surface. The case of a geometrically ruled rational surface is fully studied, e.g., the monoid of numerically effective divisor classes of such surface is explicitly determined and is minimally generated by two elements. On the other hand, as a consequence in the particular case where $X$ is a smooth rational surface with $K_{X}^{2}>0$, the following expected result holds: every fixed prime divisor of the complete linear system $|-K_X|$ is a $(-n)$-curve, for some integer $n\geq 1$.