An Update on the Classification of Rank 2 Weak Fano Threefolds
Joseph Cutrone, Nicholas Marshburn
TL;DR
This work advances the numerical and geometric classification of smooth weak Fano threefolds with Picard number $2$ and small anticanonical maps by constructing explicit geometric realizations through blowing up curves on Picard number $1$ Fano threefolds and updating the Sarkisov-link tables. It proves geometric existence for eight previously open $E1$-$E1$ cases and narrows the open set to four remaining instances, while providing corrected entries for related tables. The results solidify the status of many rank-two weak Fano threefolds within the MMP/Birational geometry framework and refine the overall table of known links, with the Appendix giving the most up-to-date data. These developments enhance the understanding of Sarkisov links and their role in the birational geometry of threefolds, with implications for broader classification problems in algebraic geometry.
Abstract
In this paper, an update on the classification of smooth weak Fano threefolds with Picard number two and small anti-canonical maps is given. Geometric constructions are provided for previously open numerical cases by blowing up certain curves on smooth Fano threefolds of Picard number one. This paper provides updated tables in the Appendix and reduces the 14 remaining E1-E* open cases to four.