67 problems

Kuznetsov's conjecture. The fourfold $X$ is rational if and only if … for an ordinary, untwisted projective K3 surface $S$.

Let $X$ be a smooth cubic fourfold. Write … where $\mathcal{A}_X$ is the Kuznetsov component. An associated K3 surface is a polarized K3 surface $S$ whose primitive degree-two Hodg…

Let $X$ be a smooth cubic fourfold, and let $\mathcal{C}_d$ denote the Hassett divisor parametrizing special cubic fourfolds with a labelling of discriminant $d$. The numerical con…

Let $Y$ be a cubic fourfold in the Hassett divisor $\mathcal C_{74}$. The Fano variety of lines of a general such $Y$ has no extremal ray. Rationality conjecture. Cubic fourfolds…

Let $X$ be a smooth cubic fourfold, let $F(X)$ be its Fano variety of lines, and let $S$ be a K3 surface. Galkin–Shinder's conjecture. The cubic fourfold $X$ is rational if and onl…

Non-rationality conjecture. A generic cubic fourfold is non-rational.

Let $X$ and $X'$ be smooth cubic fourfolds, and let $F(X)$ and $F(X')$ be their Fano varieties of lines. Suppose that $X$ and $X'$ are Fourier–Mukai partners, meaning that their Ku…

Let $X$ and $Y$ be smooth cubic fourfolds. They are Fourier–Mukai partners when their Kuznetsov components are equivalent by a Fourier–Mukai transform, so that…

Let $X$ and $Y$ be cubic fourfolds over the complex numbers, with Kuznetsov components $\mathcal{A}_X$ and $\mathcal{A}_Y$ defined by … The classical Derived Torelli Theorem for K3…

Hassett's period-map conjecture. The image of the period map is the complement of two specific Noether–Lefschetz divisors.

Hassett's rationality conjecture. The condition $X\in\mathcal{C}_d$ is equivalent to $X$ being rational.

Let $X,X'\subset\mathbb{P}^5$ be smooth cubic fourfolds, and let $\mathcal{A}_X$ and $\mathcal{A}_{X'}$ be their K3 categories. Kuznetsov's birationality conjecture. If there exist…

Let $f: \mathcal{S} \to B$ be a smooth projective family of complex surfaces over an irreducible base. For each fiber $S_b$, let $\mathcal{T}_{S_b}$ denote its integral polarized t…

Let $X\subset\mathbb P^5$ be a smooth cubic fourfold, let $\mathcal A_X$ be its K3 category, and let $\operatorname{Aut}_{\rm s}(\mathcal A_X)$ denote the group of symplectic autoe…

Let $X$ be a cubic fourfold and let $F(X)$ be its Fano variety of lines. Consider a polarized automorphism of $F(X)$ of odd prime order whose fixed locus is two-dimensional. A gene…

Let $X$ and $Y$ be smooth cubic fourfolds. They are Fourier–Mukai partners if there is an equivalence between their Kuznetsov components, $\mathcal{A}_X\simeq\mathcal{A}_Y$, realiz…

Let $X \subset \mathbb{P}^5$ be a smooth cubic fourfold with Kuznetsov component $\mathcal{A}_X$, and let $\widetilde{X} \to X$ be a birational modification obtained by a sequence…

Let $X \subset \mathbb{P}^5$ be a smooth cubic fourfold, and let its K3 Hodge atom be the Kuznetsov component $\mathcal{A}_X$ of $D^b\operatorname{Coh}(X)$. Interpreting the semior…

A variety is said to be of CM type when it has the complex multiplication property described in the surrounding theory. Piatetski-Shapiro–Shafarevich conjecture. Varieties of CM ty…

Let $X$ and $Y$ be cubic fourfolds over the complex numbers. Their Kuznetsov components are the admissible subcategories $\mathcal{A}_X$ and $\mathcal{A}_Y$ in the semiorthogonal d…

Let $Y$ and $Y'$ be cubic fourfolds. Write $\mathcal{A}_Y$ for the Kuznetsov component of $Y$, defined by … The cubic fourfolds are FM partners if there is a Fourier–Mukai equivale…

Let $X$ be a smooth cubic fourfold, that is, a smooth hypersurface of degree $3$ in projective space $\mathbb{P}^5$. A cubic fourfold is very general if it lies outside a countable…

Let $X$ and $Y$ be smooth cubic fourfolds, and write $\widetilde{\mathrm{H}}(X)$ and $\widetilde{\mathrm{H}}(Y)$ for the Addington–Thomas Hodge structures associated with their Kuz…

Let $X$ be a smooth cubic fourfold, and let $S$ be a K3 surface. Say that $X$ has a cohomologically associated K3 surface $S$ when the middle cohomology of $X$, as a Hodge structur…

Cubic-fourfold rationality conjecture. A smooth cubic fourfold is rational if and only if it has an associated K3 surface in the Hodge-theoretic or derived-categorical sense.