84 problems

A sextic double solid is a double cover of $\mathbb{P}^3$ branched over a sextic surface; it is factorial when its local divisor class groups are generated by Cartier divisors, and…

Hilbert-scheme genus conjecture. The variety $\Sigma(\mathcal{X})$ is a smooth curve of genus $3$ if and only if $\operatorname{Aut}(\mathcal{X})$ is finite.

Kuznetsov's conjecture. There is a correspondence

Generic non-jumping-line conjecture. The generic line is not jumping.

Modularity conjecture. The counting matrix $a_{ij}$ of a generic Fano threefold in the Iskovskikh family with parameters $(N,d)$ is in the same class as the corresponding matrix in…

Let $X$ be the weighted Fano threefold hypersurface considered in the paper, with numerical parameter $N$, and let $a_{2}$ and $a_{3}$ be the weights defining the weighted projecti…

Golyshev's conjecture. The family of counting equations for $X$ contains an equation

Let $X$ and $X'$ be the Fano threefolds under consideration in the paper. Birational isomorphism conjecture. The varieties $X$ and $X'$ can be birationally isomorphic. The source g…

cDV smoothing conjecture. $V$ has a smoothing by a small deformation. In particular,

Takagi's conjecture. The bound

Fano–Iskovskikh conjecture.

Birational distinction conjecture. The category $\mathcal{A}_X$ allows one to distinguish the birational type of $X$, or equivalently, the category $\mathcal{A}_Y$ allows one to di…

Moduli-space conjecture. The conclusion of the theorem is true for any smooth three-dimensional linear section

Simultaneous $\mathbb Q$-smoothing conjecture. The pair $S\subset V$ has a simultaneous $\mathbb Q$-smoothing.

Let $V$ be a ${\mathbb Q}$-Fano $3$-fold of Fano index $1$ and Cartier index $m$, and let $H$ be a Cartier divisor such that $m(K_V+H)\sim 0$. In this situation, the index conjectu…

Existence conjecture. All the types of curves, and hence all the corresponding types of rank-two bundles, from the two lists exist.

Many Fano 3-fold codimension 2 weighted complete intersections are birational to special Fano 3-fold hypersurfaces. The birational-model conjecture. A statement analogous to the pa…

A quasismooth member is a quasismooth Fano 3-fold weighted hypersurface belonging to one of the 95 known families of Fano 3-fold weighted hypersurfaces. The uniqueness conjecture.…

Let $X$ be a Fano threefold with terminal singularities. A Q-smoothing of $X$ is a Q-Gorenstein family $\mathcal{X}\rightarrow C\ni 0$ such that … and $\mathcal{X}_t$ has quotient…

For each of the seven specified Noether--Lefschetz divisors in the degree-$22$ moduli space, let $X$ be the Gorenstein canonical Fano threefold constructed from a general polarized…

Let $(S,L)$ be an ADE K3 surface in the degree-$22$ moduli space $\text{cusefont{U}{dutchcal}{m}{n}F}_{22}$, and let $\text{cusefont{U}{dutchcal}{m}{n}P}^{\mathrm{plt}}$ denote the…

Exceptional-collection and Landau–Ginzburg correspondence conjecture. The maximal length of an exceptional collection on $X$ equals the number of fibers with ordinary double points…

Let $X$ be the Fano threefold under consideration, let $E$ be a general Gieseker semistable rank two sheaf on $X$ with sufficiently large $c_2(E)$, and let $\mathcal U$ denote the…

Let $\mathscr{X}\rightarrow U$ be the family of Fano threefolds of Family textnumero 2.16, let $\lambda_{\operatorname{CM}}$ be its CM line bundle, and let $U^{\operatorname{ss}}$…

Let $X$ be a K-semistable $\mathbb{Q}$-Gorenstein degeneration of a smooth Fano threefold of Family textnumero 2.16. Degeneration conjecture. $X$ is isomorphic to the blow-up of a…