Positivity of divisors on blown-up projective spaces, II

TL;DR

This paper investigates the positivity properties of divisors on blow-ups of projective spaces, proving semi-ampleness and nefness results, and confirming the abundance conjecture for certain pairs.

Contribution

It constructs log resolutions and demonstrates semi-ampleness and nefness of divisors, advancing understanding of positivity in algebraic geometry.

Findings

Semi-ampleness of strict transforms on blow-ups

Nefness of F-nef divisors on moduli spaces

Validation of the abundance conjecture for specific pairs

Abstract

We construct log resolutions of pairs on the blow-up of the projective space in an arbitrary number of general points and we discuss the semi-ampleness of the strict transforms. As an application we prove that the abundance conjecture holds for an infinite family of such pairs. For $n+2$ points, these strict transforms are F-nef divisors on the moduli space $\overline{\mathcal{M}}_{0,n+3}$ in a Kapranov's model: we show that all of them are nef.