| $d$ | $({*}{*}')$ | $({*}{*})$ | $({*}{*}{*})$ | $({*}{*}'')$ |
|---|
References
Explanation
A smooth cubic fourfold $X\subseteq\mathbb{P}^5$ is special of discriminant $d$ if it contains an algebraic surface $S$ such that $\langle h^2,S\rangle\subseteq\operatorname{H}^{2,2}(X)\cap\operatorname{H}^4(X,\mathbb{Z})$ is a saturated sublattice of rank 2 and discriminant $d$, where $h$ denotes the hyperplane class. Inside the moduli space $\mathcal{C}$ of smooth cubic fourfolds the special cubic fourfolds of discriminant $d$ form an irreducible divisor $\mathcal{C}_d$, introduced in Hassett (2000).
- $({*})$
Hassett's condition for non-emptiness of $\mathcal{C}_d$
$d$ satisfied $d\geq 8$ and $d\equiv 0,2\pmod 6$
introduced in Hassett (2000)
- $({*}{*}')$
Huybrechts' condition for an associated K3 twisted surface
$d$ satisfies $({*})$ and in the prime factorisation $d/2=\prod p_i^{n_i}$ one has $n_i\equiv0\pmod2$ for all primes $p_i\equiv2\pmod3$
equivalently, $d=2(a^2+ab+b^2)$ for some $a,b\in\mathbb{Z}$
there exists a K3 surface $S$ and a Brauer class $\alpha\in\operatorname{Br}(S)$ such that $\widetilde{\operatorname{H}}(X,\mathbb{Z})\cong\widetilde{\operatorname{H}}(S,\alpha,\mathbb{Z})$
introduced in Huybrechts (2017)
- $({*}{*})$
Hassett's condition for an associated K3 surface
$d$ satisfies $({*})$ and $d$ is not divisible by $4$, $9$ or any odd prime $p\equiv2\pmod3$
equivalently, $d=2(a^2+ab+b^2)$ for some coprime $a,b\in\mathbb{Z}$
there exists a K3 surface $S$ such that $\widetilde{\operatorname{H}}(X,\mathbb{Z})\cong\widetilde{\operatorname{H}}(S,\mathbb{Z})$
there exists a K3 surface $S$ such that $S$ is birational to a moduli space of stable sheaves on $S$
introduced in Hassett (2000)
- $({*}{*}{*})$
Addington's condition for an associated K3 surface
$d$ satisfies $({*})$ and $d$ is of the form $\displaystyle\frac{2n^2+2n+2}{a^2}$ for some $n,a\in\mathbb{Z}$
there exists a K3 surface $S$ such that $\mathrm{F}(X)$ is birational to $\operatorname{Hilb}^2S$
introduced in Addington (2016)
- $({*}{*}'')$
condition for a rational Lagrangian fibration
$d$ satisfies $({*})$ and in the prime factorisation $d/2=\prod p_i^{n_i}$ one has $n_i\equiv0\pmod2$ for all primes $p_i$
equivalently, $d=2a^2$ for some $a\in\mathbb{Z}$
the Fano variety of lines $\mathrm{F}(X)$ admits a rational Lagrangian fibration
featured in Bottini–Huybrechts (2025)
We have that \[ ({*}{*}{*})\Rightarrow({*}{*})\Rightarrow({*}{*}')\Leftarrow({*}{*}'') \] where $({*}{*}'')$ is incomparable with $({*}{*})$ and $({*}{*}{*})$.
In the column for the degree of the (twisted) K3 surface, an entry $e\ (k)$ denotes the degree of the K3 surface (the order of the Brauer class).
The column $\#\operatorname{FM}$ gives the number of isomorphism classes of Fourier–Mukai partners of the K3 category of a very general element of $\mathcal{C}_d$.
A checkmark in the column rational indicates that every smooth cubic fourfold in $\mathcal{C}_d$ is known to be rational. Conjecturally these are precisely the ones satisfying $({*}{*})$.
The column contains describes a surface $S$ contained in the generic element of $\mathcal{C}_d$, together with the pair $(h^2\cdot S, S^2)$ giving its degree and self-intersection. Here $\operatorname{Bl}_p\mathbb{P}^2$ denotes the blowup of $\mathbb{P}^2$ in $p$ generic points, embedded by an explicit linear system as in Nuer's table of smooth rational surfaces.
About
This is an overview of (some of) what we know about the relationship between cubic fourfolds and K3 surfaces.
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