25 problems

Grinenko's conjecture. $X$ is birationally rigid if and only if it satisfies the $K$-condition.

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 $F_1,\dots,F_K$ be primitive Fano varieties, meaning Fano varieties in the class considered in the paper, and let … Assume that each $F_i$ is birationally (super)rigid. Direct-…

Let $V\to\mathbb{P}^1$ be a Fano fibration satisfying the $K$-condition, which is weaker than the $K^2$-condition. K-condition conjecture. The $K$-condition is sufficient for birat…

Let $V\to S$ be a fibration into Fano varieties satisfying (i) $V$ is smooth and $\mathop{\rm dim}V\geq 4$; (ii) $\mathop{\rm dim}S\geq 1$ and the anticanonical class $-K_V$ is rel…

Conjecture 2. The variety $V$ is birationally rigid. The conjecture weakens the smoothness assumption in the preceding absolute statement; the higher-dimensional birational rigidit…

Conjecture 1. The variety $V$ is birationally superrigid. This extends the classes of higher-dimensional Fano varieties for which birational superrigidity is known, including certa…

Twisted fibration rigidity conjecture. If the fibration $V/S$ as above is sufficiently twisted over the base $S$, then $V/S$ is birationally rigid.

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

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 quasismooth Fano threefold weighted hypersurface with Fano index $iota_X=1$, and let $X$ be any quasismooth member of its deformation family. Corti–Pukhlikov–Reid conj…

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.

Birational rigidity conjecture. The fibre space $V\to S$ is birationally rigid, respectively birationally superrigid, and every birational map from its total space to the total spa…

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…

Iskovskikh's conjecture. If

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.

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.