Gushel–Mukai 4-folds

The $\kappa(\mathcal{GM})$ column concerns the $23$-dimensional GM component; $\kappa(\mathcal D)$ concerns its $19$-dimensional period divisor. They are separate invariants because the GM period map has $4$-dimensional fibers. In the K3 column, $e\ (r)$ means degree $e$ and Brauer order $r$. A Fourier–Mukai entry $n$ is exact in the untwisted case; $\geq n$ is the strongest lower bound supplied by the displayed twisted presentations. On a primed or double-primed row, the parenthesized number is the total over the two components; the first number is the count in that component. The fibers counted may be empty because the image of the GM period map is not completely known. In the *contains* column, $(e,g)$ is the degree and sectional genus of a rational surface in the generic ordinary fourfold. An em dash means “not applicable”; a question mark means that no value or generic surface model is known.
$d$	$(\ast\ast^{\prime})$	$(\ast\ast)$	$(\ast\ast\ast)$

References

Explanation

A smooth Gushel–Mukai fourfold $X$ is Hodge-special if the integral Hodge classes in $\mathrm{H}^4(X,\mathbb{Z})$ contain a primitive rank-$3$ lattice $L_d$ containing the rank-$2$ lattice pulled back from $\operatorname{Gr}(2,5)$. The discriminant of $L_d$ is denoted by $d$. The period map from the $24$-dimensional moduli space of GM fourfolds to the $20$-dimensional period domain is dominant with smooth $4$-dimensional fibers, and the Hodge-special locus is the pullback of a union of Noether–Lefschetz divisors, as developed in Debarre–Iliev–Manivel (2015).

The possible discriminants satisfy:

\begin{equation} (\ast)\qquad d>8 \quad\text{and}\quad d\equiv0,2,4\pmod 8. \end{equation}

If $d\equiv0$ or $4\pmod8$, the divisor $\mathcal{GM}_d$ is irreducible. If $d\equiv2\pmod8$, it has two irreducible components $\mathcal{GM}_d'$ and $\mathcal{GM}_d''$, which appear as separate rows.

$(\ast\ast^{\prime})$

Pertusi's condition for an associated twisted K3 surface.

$d$ satisfies $(\ast)$, and in the prime factorization $d=\prod p_i^{n_i}$ one has $n_i\equiv0\pmod2$ for every prime $p_i\equiv3\pmod4$.

There is a K3 surface $S$ and a Brauer class $\alpha\in\operatorname{Br}(S)$ such that the Mukai lattice of $\operatorname{Ku}(X)$ is Hodge isometric to $\widetilde{\mathrm{H}}(S,\alpha,\mathbb{Z})$.

Introduced in Pertusi (2019).

$(\ast\ast)$

Debarre–Iliev–Manivel's condition for a Hodge-associated K3 surface.

$d$ satisfies $(\ast)$, is not divisible by $8$, and every odd prime dividing $d$ is congruent to $1\pmod4$.

Equivalently, $d=2(a^2+b^2)$ for some coprime integers $a,b$.

There is a degree-$d$ polarized K3 surface $S$ such that the primitive cohomology of $S$ and the vanishing cohomology of $X$ are Hodge isometric up to sign and Tate twist.

Introduced in Debarre–Iliev–Manivel (2015).

$(\ast\ast\ast)$

Pertusi's condition for the associated double EPW sextic to be birational to a Hilbert square of a K3 surface.

$d$ satisfies $(\ast)$ and \begin{equation} a^2d=2n^2+2 \end{equation} for some integers $a,n$; equivalently, the negative Pell equation $n^2-(d/2)a^2=-1$ has an integral solution.

When the associated double EPW sextic $\widetilde{Y}_A$ is smooth, there is a K3 surface $S$ such that $\widetilde{Y}_A$ is birational to $\operatorname{Hilb}^2(S)$.

Introduced in Pertusi (2019).

We have \begin{equation} (\ast\ast\ast)\Longrightarrow(\ast\ast)\Longrightarrow(\ast\ast^{\prime}). \end{equation}

The arithmetic entries in the K3 and Fourier–Mukai columns are computed from Brakkee–Pertusi (2021).

About

This is an extension of the overview of relationships between special cubic fourfolds and K3 surfaces to Noether–Lefschetz divisors for Gushel–Mukai fourfolds.

See the GitHub repository for more information and feature requests.