Towards the Classification of Weak Fano Threefolds with \rho = 2

TL;DR

This paper classifies and provides examples of weak Fano threefolds with Picard number 2, focusing on Sarkisov links and extremal rays, advancing understanding of their geometric structures.

Contribution

It offers a numerical classification of smooth weak Fano threefolds with a=2 and extremal rays of type E, including existence proofs for some cases.

Findings

Examples of Sarkisov links of type II between Fano threefolds with a=1.

Numerical classification of weak Fano threefolds with a=2 and extremal rays.

Proof of existence for certain classified cases.

Abstract

In this paper, we provide examples of Sarkisov links of type II between complex projective Fano threefolds $X$ with $\rho(X) = 1$. To show examples of these links, we study smooth weak Fano threefolds with extremal rays of type $E$. We further assume that the pluri-anticanonical morphism contracts only a finite number of curves. We numerically classify smooth weak Fano threefolds with $\rho(X) = 2$ having an extremal ray of type $E$ both before and after a flop as well as prove the existence of some of these numerical cases.