The blow-up of ℙ⁴ at 8 points and its Fano model, via vector bundles on a del Pezzo surface

2017/07/28 by Casagrande, C., Codogni, G., Fanelli, A.
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1707.09152

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 c₁=-K_S and c₂=2, on a polarized degree one del Pezzo surface (S,L); this is related to the birational geometry of the blow-up X of P⁴ 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,-KS, 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.

Related

• Anticanonical geometry of the blow-up of ℙ⁴ in 8 points and its Fano model
• Blowups, Gale duality, and moduli spaces
• Birational geometry of moduli space of del Pezzo pairs
• Moduli spaces of sheaves on Fano threefolds and K3 surfaces of genus 9
• Espace des modules des faisceaux semi-stables de rang 2 et de classes de Chern c₁=0, c₂=2 et c₃=0 sur une hypersurface cubique lisse de ℙ⁴