23 problems

Iskovskikh's conjecture. The base surface $Z$ has only canonical singularities, or equivalently $Z$ is $1$-lc.

Let $S$ be a rational surface, let $\Delta\subset S$ be a discriminant curve, and let $\pi:X\to S$ be a standard conic bundle. Write $K_S$ for the canonical divisor of $S$,…

Let $V$ be the double-cover variety of type $((0),1)$, with conic-bundle projection $q\colon V\to{\mathbb P}$. Let $\operatorname{Bir}(V/{\mathbb P})$ denote the group of fiber-wis…

Let $G\subset{\mathbb P}$ be general, let $F\to G$ be the associated Fano double cover, and set $V\cong F\times{\mathbb P}^1$. The projection $q\colon V\to F$ is a conic-bundle str…

Let $G\subset{\mathbb P}$ be a general hypersurface and let $V$ be the associated double-cover variety. The projection $q_V\colon V\to G$ is a conic bundle, and…

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…

Log-terminality conjecture. The divisor pair

The log-terminal anticanonical divisor conjecture. For every $n\in\mathbb{N}$, there exists a divisor $D\in |-nK_X|$ such that

The $A_n$-base conjecture. If $f\colon (X,C)\to (S,0)$ is a Mori conic bundle, then $(S,0)$ is a Du Val singularity of type $A_n$.

Spinor-modification hyperbolic-equivalence conjecture. If $X'/S$ is a spinor modification of $X/S$, then $X'/S$ is hyperbolic equivalent to $X/S$.

Let $\pi\colon X\to W$ be a conic bundle threefold over a geometrically rational smooth surface $W$, defined over a field $k$ of characteristic different from $2$. Folklore conject…

Let $\pi:X\to S$ be a standard conic bundle, and suppose that there is a birational map … A Cremona transformation is a birational self-map of $\mathbb{P}^3$. Kantor's conjecture.…

Let $k$ be a number field, let $\pi:X\to\mathbb{P}^n$ be a conic bundle, and let $F$ be the pullback of the hyperplane class. Let $\alpha\in\mathbb{Q}_{>0}$ be such that…

Let $\pi \colon X \to S$ be an $n$-dimensional standard conic bundle with $n \ge 3$, and let $\Delta$ denote the discriminant divisor of $\pi$. Higher-dimensional discriminant conj…

Shokurov's conjecture. If

Manin's rationality conjecture. Rationality is almost equivalent to

Sarkisov phantom-category conjecture. The semiorthogonal decomposition of $D^b(X)$ contains a phantom category.

Let $T$ be a 3-fold and let $\pi:T\to W$ be a standard conic bundle, with connected discriminant curve $\Delta\subset W$. Let $\Lambda$ be a base point free pencil of rational curv…

Let $X$ be a conic bundle of dimension $3$. Let $D^b(X)$ denote its bounded derived category, and suppose … is a semiorthogonal decomposition into a phantom category $\mathcal A$ a…

Let $X$ be the Brauer–Severi variety of an order of type $F^2_3$, $F^2_4$, or $F^{2-}_5$, as described in the table in the source. Let $Y$ be another variety birational to $X$ over…

Let $C$ be the canonically embedded complete-intersection curve of degree $8$ and genus $5$ arising as the base locus of a net of quadrics in $\mathbb{P}^4$, let…

Let $C$ be the canonically embedded complete-intersection curve of degree $8$ and genus $5$ arising as the base locus of a net of quadrics in $\mathbb{P}^4$. Fix ramification data…