15 problems

Semistable dP3 rigidity conjecture. Then $X/\mathbb{P}^1$ is birationally rigid. This is a semistable variant of Grinenko's conjecture.

Grinenko's dP3 rigidity conjecture. If $X$ satisfies condition $(\ast)$, then $X$ is birationally rigid. This conjecture extends the known rigidity theorem under the stronger $K^2$…

Let $\pi\colon X\to\mathbb{P}^2$ be a standard conic bundle, and let its discriminant curve have degree $d$. High-degree discriminant rigidity conjecture. If $d\geq 9$, then…

Standard conic bundle rigidity conjecture. If $X\to\mathbb{P}^2$ satisfies condition $(\ast)$, then it is birationally rigid. This is proposed as a criterion for birational rigidit…

Let $X/S$ and $Y/T$ be threefold Mori fibre spaces, and let rigidity mean birational rigidity of the Mori fibre space. Topological rigidity conjecture. Whether a threefold Mori fib…

Classification conjecture. Any smooth Mori fibration birational to $U$ is biregular to either $U$ or $V_l$ for some curve $l$.

Let $X$ be a $\mathbb Q$-factorial sextic double solid, meaning a double cover of $\mathbb P^3$ branched over a sextic surface, with singularities of the indicated types. Sextic doub…

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…

Let $X$ be a quasismooth Fano threefold weighted complete intersection of codimension $2$ and index $1$, belonging to a deformation family indexed by $I$. Let $I_{br}$, $I_F$, and…

Non-rigidity and non-solidity conjecture. Then $X$ is not birationally rigid. Moreover, if the codimension is at least $3$, then $X$ is non-solid.

Let $d$ be an integer with $1\leq d\leq 3$. Let $X\to\mathbb P^1$ be a $\mathfrak D$-semistable del Pezzo fibration of degree $d$, and assume that it satisfies the $K$-condition, m…

Let $X\to\mathbb{P}^1$ be a del Pezzo fibration of degree $1$ with at most terminal singularities. It satisfies the $K$-condition when $-K_X$ is not in the interior of the movable…

A del Pezzo fibration is birationally rigid if it has only one Mori fiber space structure in its birational class. Let $X$ be a del Pezzo fibration satisfying the hypotheses of The…

Semistability-modified conjecture. If $X$ satisfies the $K$-condition, then $X$ is birationally rigid.

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