47 problems

The maximal third Chern character conjecture. If $c=-1$ and $d=\frac12+\frac C5\ll0$, then

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

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

Fano–Iskovskikh 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 $X$ 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…

Let $\mathrm{X}_{1.9}$ and $\mathrm{X}'_{1.9}$ be the hourglass varieties, and for each $i$ let $\mathcal{MF}_{i}(\mathrm{X}_{1.9})$ denote the left orthogonal component of the exc…

Kuznetsov's conjecture. There is a correspondence

Let $X$ be a cubic threefold, quartic double solid or Gushel--Mukai threefold, and let $\mathcal{K}u(X)$ be its Kuznetsov component. Let $\sigma$ be a stability condition on…

Zinger's comparison conjecture. One has

Dolgachev–Nikulin duality conjecture. There is an isomorphism of lattices

Let $Y\to\mathbb{P}^3$ be a quartic double solid whose ramification divisor contains no lines, and let $\cX/B$ be the family of prime Fano threefolds of genus $6$ constructed in Th…

Let $X$ be the smooth Fano threefold obtained by blowing up a smooth quadric threefold $Q\subset\mathbb{P}^4$ along the smooth twisted rational quartic curve $C_4$, so that…

Let $X_d$ be a quasismooth member of one of the eight deformation families numbered textnumero 1, textnumero 19, textnumero 28, textnumero 39, textnumero 49, textnumero 59, textnum…

Mirror correspondence. There is a bijective correspondence between qG-deformation families of Q-Fano threefolds $X$ of class TG and mutation-equivalence classes of rigid maximally…

Let $X$ be a smooth Fano threefold. Write $\operatorname{Nef}_1(X)_{\mathbb{Z}}$ for the integral nef curve classes, and let $\operatorname{Mor}(\mathbb{P}^1,X,\alpha)$ denote the…

Let $Y$ be a quartic double solid and let $X$ be a GM threefold. Let $J(Y)$ and $J(X)$ denote their intermediate Jacobians, each with its principal polarization. Intermediate-Jacob…

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.

Local Fano threefold genus-zero correspondence. One has

Let $X$ and $X'$ be prime Fano threefolds of index one or two, with intermediate Jacobians $J(X)$ and $J(X')$ and Kuznetsov components $\mathcal{K}u(X)$ and $\mathcal{K}u(X')$. Cat…

Non-rationality conjecture. Then $X$ is never ${\mathsf{k}}$-rational.

Let $\mathcal{MF}^i_d$ be the moduli spaces of prime Fano threefolds of index $i$ and degree $d$. A correspondence conjecture. There is a correspondence…

K-semistability conjecture. Every smooth Fano threefold with $\rho(X)=1$ is K-semistable.