10 problems

Let $X$ be a $mathbb Q$-factorial quartic threefold, equivalently a factorial quartic threefold, whose singular points are all of type $cA_1$. Quartic cA1 rigidity conjecture. The…

Nodal hypersurface factoriality conjecture. Under any one of these three conditions, $V$ is $\mathbb{Q}$-factorial.

Birational-rigidity conjecture. The quartic threefold $V$ is birationally rigid.

Equivariant birational rigidity conjecture. Then $X$ is $G$-birationally rigid.

Toric quartic threefold deformation conjecture. These four toric varieties are the only K-polystable Fano toric threefolds that deform to quartic threefolds.

Let $X$ be a smooth quartic threefold over ${\mathbf R}$. A rational curve on $X$ has trivial Borel–Haefliger class when its Borel–Haefliger class in…

Birigidity conjecture. The variety $X$ is birigid, meaning

Pliability conjecture. The set $\mathcal P(X)$ is finite, and

Rigidity conjecture. Every terminal factorial quartic threefold with no worse than $cA_1$ points is rigid.

Let $Y_4\subset \mathbb{P}^4$ be a terminal quartic hypersurface, and let $\operatorname{Cl}Y$ denote its divisor class group. Terminal quartic pliability-or-rationality conjecture…