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

Understand

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.