  We characterise smooth curves in 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 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.
