On ruled surfaces with big anti-canonical divisor and numerically trivial divisors on weak log Fano surfaces

TL;DR

This paper studies the structure of certain ruled surfaces with big anti-canonical divisors and explores the Picard group structure of normal projective surfaces with nef and big anti-canonical classes.

Contribution

It provides new insights into the structure of ruled surfaces with big anti-canonical divisors and characterizes the Picard group of specific normal projective surfaces.

Findings

Ruled surfaces with big anti-canonical class have a specific structure.

The Picard group of a normal projective surface with nef and big anti-canonical class is free abelian of finite rank.

Abstract

We investigate the structure of geometrically ruled surfaces whose anti-canonical class is big. As an application we show that the Picard group of a normal projective surface whose anti-canonical class is nef and big is a free abelian group of finite rank.