Anticanonical geometry of the blow-up of P4 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 = MS,−KS which is the moduli space of semi-stable ranktwo torsion-free sheaves with c1 = −KS and c2 = 2 on a polarised degree-one del Pezzo surface (S, −KS). Based on the relation between Y and the blow-up of P4 in 8 points, we describe completely the base scheme of the anticanonical system |−KY |. We also prove that the Bertini involution ιY of Y , induced by the Bertini involution ιS of S, preserves every member in |−KY |. In particular, we establish the relation between ιY and the anticanonical map of Y , and we describe the action of ιY by analogy with the action of ιS on S.