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

C. Casagrande; G. Codogni; A. Fanelli

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arXiv · 1707.09152

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

C. Casagrande·G. Codogni·A. Fanelli

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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.

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C. Casagrande, G. Codogni, A. Fanelli. 2017-07-28. The blow-up of $\mathbb{P}^4$ at 8 points and its Fano model, via vector bundles on a del Pezzo surface. https://arxiv.org/abs/1707.09152

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