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

TL;DR

This paper characterizes specific curves on hyperquadrics in projective 4-space whose blowups yield weak Fano threefolds, establishing their geometric realizability and constructing related Sarkisov links.

Contribution

It provides a complete geometric classification of curves leading to weak Fano threefolds via blowups, confirming their existence beyond numerical predictions.

Findings

Characterization of curves with no 4-secant line or 7-secant conic

Verification of the existence of weak Fano threefolds from these curves

Construction of Sarkisov links from the classified threefolds

Abstract

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 $(g,d)\not \in \{(4,7),\:(10, 11)\}$; (iii) either $3d-26<g\leq\frac{d^2-1}{12}$ 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.