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

Abstract

AbstractBuilding on the work of Casagrande–Codogni–Fanelli, we develop our study on the birational geometry of the Fano fourfold $$Y=M_{S,-K_S}$$ Y = M S , - K S which is the moduli space of semi-stable rank-two torsion-free sheaves with $$c_1=-K_S$$ c 1 = - K S and $$c_2=2$$ c 2 = 2 on a polarised degree-one del Pezzo surface $$(S,-K_S)$$ ( S , - K S ) . Based on the relation between Y and the blow-up of $$\mathbb {P}^4$$ P 4 in 8 points, we describe completely the base scheme of the anticanonical system $$|{-}K_Y|$$ | - K Y | . We also prove that the Bertini involution $$\iota _Y$$ ι Y of Y, induced by the Bertini involution $$\iota _S$$ ι S of S, preserves every member in $$|{-}K_Y|$$ | - K Y | . In particular, we establish the relation between $$\iota _Y$$ ι Y and the anticanonical map of Y, and we describe the action of $$\iota _Y$$ ι Y by analogy with the action of $$\iota _S$$ ι S on S.