| $d$ | $({*}{*})$ | $({*}{*}')$ | $({*}{*}{*})$ |
|---|
References
Explanation
- $({*})$
-
Hassett's condition for non-emptiness of $\mathcal{C}_d$
$d$ satisfied $d\geq 8$ and $d\equiv 0,2\pmod 6$
- $({*}{*})$
-
Hassett's condition for an associated K3 surface
$d$ satisfies $({*})$ and $d$ is not divisible by $4$, $9$ or any odd prime $p\equiv2\pmod3$
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 $S$
- $({*}{*}')$
-
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 $p_i\equiv2\pmod3$
there exists a K3 surface $S$ and a Brauer class $\alpha\in\operatorname{Br}(S)$ such that $\widetilde{\operatorname{H}}(X,\mathbb{Z})\cong\widetilde{\operatorname{H}}(S,\alpha,\mathbb{Z})$
- $({*}{*}{*})$
-
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 $\operatorname{Hilb}^2S$
We have that \[ ({*}{*}{*})\Rightarrow({*}{*})\Rightarrow({*}{*}') \]
About
This is an overview of (some of) what we know about the relationship between cubic fourfolds and K3 surfaces.
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