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