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

TL;DR

This paper studies the birational geometry of a specific Fano fourfold related to the blow-up of projective 4-space at 8 points, describing its anticanonical system and involution symmetries.

Contribution

It provides a complete description of the base scheme of the anticanonical system and analyzes the Bertini involution's action, linking it to the involution on the underlying del Pezzo surface.

Findings

Complete description of the base scheme of |−K_Y|

Proof that the Bertini involution preserves |−K_Y|

Relation established between involutions and the anticanonical map

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 $S$.