Anticanonical geometry of the blow-up of ℙ⁴ in 8 points and its Fano model

2021/11/03 by Xie, Zhixin
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2111.02084

Abstract

Building on the work of Casagrande-Codogni-Fanelli, we develop our study on the birational geometry of the Fano fourfold Y=M_S,-KS which is the moduli space of semi-stable rank-two torsion-free sheaves with c₁=-K_S and c₂=2 on a polarised degree-one del Pezzo surface (S,-K_S). 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 |-K_Y|. We also prove that the Bertini involution ι_Y of Y, induced by the Bertini involution ι_S of S, preserves every member in |-K_Y|. In particular, we establish the relation between ι_Y and the anticanonical map of Y, and we describe the action of ι_Y by analogy with the action of ι_S on S.

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• Classification of Fano fourfolds with large anticanonical base locus
• An example of Mirror Symmetry for Fano threefolds
• Footnotes to a footnote of a paper by Fano
• K-moduli of Fano threefolds and genus four curves