Weak Fano threefolds arising as the blowup of a hyperquadric in ${\mathbb{P}}^4$ along a curve

We characterize smooth irreducible curves $C$ on a smooth hyperquadric $Y$ of ${\mathbb{P}}^4$ such that the blowup of $Y$ along $C$ is a weak Fano threefold. These are precisely the smooth irreducible curves $C$ of degree $d$ and genus $g$ lying on a smooth hypercubic section of $Y$ such that (i) $C$ has no 4-secant line and no 7-secant conic; (ii) $d<18$ and $(g,d)\in ̸\{(4,7),(10,11)\}$; (iii) either $3d-26<g\le \frac{d^2-1}{12}$ or $(g,d)\in \{(4,6),(13,12)\}$. We prove the geometric realizability of each case, thereby proving the existence of weak Fano threefolds and Sarkisov links constructed from them, which were previously known only as numerical possibilities.