Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links

TL;DR

This paper characterizes certain smooth curves in P^3 whose blow-ups yield weak Fano threefolds with big and nef anticanonical divisors, and constructs Sarkisov links from these blow-ups, confirming their geometric existence.

Contribution

It provides a classification of curves leading to weak Fano threefolds via blow-ups and explicitly constructs Sarkisov links previously known only numerically.

Findings

Characterization of curves in P^3 with big and nef anticanonical divisors after blow-up.

Explicit construction of Sarkisov links from these blow-ups.

Confirmation of the geometric existence of certain Sarkisov links.

Abstract

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.