Anticanonical geometry of the blow-up of mathbb {P}^4 in 8 points and its Fano model

Abstract

Building on the work of Casagrande–Codogni–Fanelli, we develop our study on the birational geometry of the Fano fourfold Y=M_{S,-K_S} which is the moduli space of semi-stable rank-two torsion-free sheaves with c_1=-K_S and c_2=2 on a polarised degree-one del Pezzo surface (S,-K_S). Based on the relation between Y and the blow-up of mathbb {P}^4 in 8 points, we describe completely the base scheme of the anticanonical system |{-}K_Y|. We also prove that the Bertini involution iota _Y of Y, induced by the Bertini involution iota _S of S, preserves every member in |{-}K_Y|. In particular, we establish the relation between iota _Y and the anticanonical map of Y, and we describe the action of iota _Y by analogy with the action of iota _S on S.