3-Fold Links with Smooth Central Models

We are classifying Sarkisov links with smooth centers of dimension 3. These centers are weak Fano varieties with Picard number 2 whose anti-canonical divisor is big in codimension 1. Takeuchi geometrically classified those where one side of the link is a del Pezzo fibration of degree not equal to 6. Jahnke, Peternell, and Radloff gave examples when one side of the link is a conic bundle. Classical examples of the remaining divisorial type were given by Iskovskikh and Fano and more recently by Takeuchi. We will report on our progress to classify numerically these last types as well as their geometric realizability.