Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links
Résumé
We characterise smooth curves in $\mathbb{P}^3$ whose blow-up produces a threefold with anticanonical divisor big and nef. These are curves $C$ of degree $d$ and genus $g$ lying on a smooth quartic, such that (i) $4d-30 \le g\le 14$ or $(g,d) = (19,12)$, (ii) there is no 5-secant line, 9-secant conic, nor 13-secant twisted cubic to $C$. This generalises the classical similar situation for the blow-up of points in $\mathbb{P}^2$. We describe then Sarkisov links constructed from these blow-ups, and are able to prove the existence of Sarkisov links which were previously only known as numerical possibilities.