The blow-up of $\mathbb{P}^4$ at 8 points and its Fano model, via vector bundles on a del Pezzo surface

TL;DR

This paper investigates the birational geometry of a specific Fano 4-fold obtained from the blow-up of P^4 at 8 points, using moduli spaces of sheaves on a del Pezzo surface and stability conditions.

Contribution

It provides an explicit analysis of the Fano model via vector bundles on a del Pezzo surface, detailing divisor cones, automorphisms, and linear systems.

Findings

Description of the divisor cones of the Fano 4-fold

Analysis of the automorphism group of the Fano model

Investigation of the base loci of key linear systems

Abstract

Building on the work of Mukai, we explore the birational geometry of the moduli spaces M_{S,L} of semistable rank two torsion-free sheaves, with c_1=-K_S and c_2=2, on a polarized degree one del Pezzo surface (S,L); this is related to the birational geometry of the blow-up X of P^4 in 8 points. Our analysis is explicit and is obtained by looking at the variation of stability conditions. Then we provide a careful investigation of the blow-up X and of the moduli space Y=M_{S,-K_S}, which is a remarkable family of smooth Fano 4-folds. In particular we describe the relevant cones of divisors of Y, the group of automorphisms, and the base loci of the anticanonical and bianticanonical linear systems.