An Update on the Classification of Rank 2 Weak Fano Threefolds

TL;DR

This paper updates the classification of smooth weak Fano threefolds with Picard number two, providing new geometric constructions and reducing unresolved cases from fourteen to four.

Contribution

It offers an updated classification with explicit geometric constructions and narrows down remaining open cases in the classification of weak Fano threefolds.

Findings

Reduced the number of open cases from 14 to 4.

Provided explicit geometric constructions for previously unresolved cases.

Updated classification tables for weak Fano threefolds.

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.