Explicit log Fano structures on blow-ups of projective spaces

TL;DR

This paper classifies blow-ups of projective spaces at general points that admit explicit log Fano structures, providing concrete boundary divisors to establish their log Fano property.

Contribution

It explicitly determines which blow-ups of projective spaces are log Fano and constructs explicit boundary divisors for these cases.

Findings

Identifies conditions for blow-ups of projective spaces to be log Fano.

Provides explicit boundary divisors for the log Fano structures.

Enhances understanding of the geometry of blow-ups and their Fano properties.

Abstract

In this paper we determine which blow-ups $X$ of $\mathbb{P}^n$ at general points are log Fano, that is, when there exists an effective $\mathbb{Q}$-divisor $\Delta$ such that $-(K_X+\Delta)$ is ample and the pair $(X,\Delta)$ is klt. For these blow-ups, we produce explicit boundary divisors $\Delta$ making $X$ log Fano.