Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-90-25-d

·

Blow-up of a hyperplane section of the adjoint $G_2$ fivefold along a K3 surface.

  • $\pi_1\colon X\to W$ is the blow-up of the Mukai fourfold $W$ of genus $10$, a smooth hyperplane section of the adjoint $G_2$-variety in $\mathbb P^{13}$ (1-288-63-a), along a K3 surface of degree $18$ and genus $10$, a codimension-two linear section of $W$.

  • $\pi_2\colon X\to\mathbb P^1$ is a fibration whose general fiber is a smooth prime Fano threefold of genus $10$ and degree $18$; for general $X$ there are $30$ singular fibers, each with exactly one ordinary double point.

This is a Fano fourfold of K3 type, appearing as case K3-28 in Bernardara–Fatighenti–Manivel–Tanturri.

Varieties in this family can be blown down to

Defining data
Ambient
ℙ1P1 × Gr(2, 7)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(w3) + E^(w4) + E^(w1 + w3)
Bundle
O_2(1) + Q_2^v(1) + O_1(1) box O_2(1)
ZL62 label
1601.32w1
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;7) \\ \det\mathcal{E} &\cong \mathcal{O}(1;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 25
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 90
$\chi(X, \mathrm{T}_X)$ = -21

$\omega_X^\vee \cong \mathcal{O}_X(1)$

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 90
$\mathrm{c}_1^2\mathrm{c}_2$ = 108
$\mathrm{c}_2^2$ = 128
$\mathrm{c}_1\mathrm{c}_3$ = 24
$\mathrm{c}_4$ = 30
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 22 1 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x2 + 21 0 x0 + 21 x2
$\mathrm{HH}^{ 3 }$ = x1 + 3 0 0 x1 3
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 11 0 0 x1 - x2 - 14 0 25
$\mathrm{HH}^{ 5 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 6 }$ = 0 0 0 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 20t + 48t^{2} + 20t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) No match
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) No match

Quantum differential equation

With $D=t\frac{\mathrm{d}}{\mathrm{d}t}$, the regularized period satisfies $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 3t + 1 \right) \left( 4t + 1 \right) ^{3} \left( 4752t^{4} + 864t^{3} - 72t^{2} - 24t + 1 \right) \left( 40979568t^{6} - 885024t^{5} - 7406212t^{4} - 1553412t^{3} - 46448t^{2} + 1322t - 5 \right)D^{6} + 2 \left( 4t + 1 \right) ^{2} \left( 24536598299136t^{12} + 13182518845440t^{11} - 3278572578432t^{10} - 3967343962560t^{9} - 1078952699520t^{8} - 102445996224t^{7} + 2407542024t^{6} + 805737348t^{5} + 22559370t^{4} + 2393290t^{3} + 99349t^{2} - 1948t + 5 \right)D^{5} + \left( 4t + 1 \right) \left( 1635773219942400t^{13} + 1017815941094400t^{12} - 163576648772352t^{11} - 308596227780672t^{10} - 107730939066048t^{9} - 16673360340432t^{8} - 1024709463552t^{7} + 10992328288t^{6} + 2776526068t^{5} - 34933372t^{4} - 12689624t^{3} - 197322t^{2} + 2171t - 5 \right)D^{4} + 2 t \left( 13740495047516160t^{13} + 10203028773396480t^{12} - 531576904435200t^{11} - 2983756628344320t^{10} - 1328515238910528t^{9} - 277974647299008t^{8} - 29853629733168t^{7} - 1377812597760t^{6} + 2799696456t^{5} + 1571305144t^{4} - 3273082t^{3} - 836226t^{2} + 29915t + 9 \right)D^{3} + 4 t ^{2} \left( 15179975481065472t^{12} + 9765915460955136t^{11} - 1534490990896896t^{10} - 3253191845200704t^{9} - 1277164084716432t^{8} - 241417697573232t^{7} - 23411248084500t^{6} - 984186886332t^{5} - 775708556t^{4} + 809164296t^{3} + 447079t^{2} + 203489t + 9415 \right)D^{2} + 48 t ^{2} \left( 1374049504751616t^{12} + 784086167950848t^{11} - 194955992424576t^{10} - 288108777746496t^{9} - 103055298779184t^{8} - 17916876042592t^{7} - 1584440335276t^{6} - 59601154472t^{5} + 60052322t^{4} + 47020578t^{3} + 64222t^{2} + 24089t + 260 \right)D + 48 t ^{2} \left( 560836532551680t^{12} + 292488032471040t^{11} - 93893427224064t^{10} - 115284978636480t^{9} - 38629594139184t^{8} - 6300016687056t^{7} - 516351788604t^{6} - 17424436356t^{5} + 77895232t^{4} + 14845176t^{3} + 66349t^{2} + 7727t + 40 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × Gr(2, 7)

|λ| = 25.19964 λ ≈ 25.19964; multiplicity 1; Jordan blocks 1 λ ≈ 7.584966; multiplicity 1; Jordan blocks 1 λ ≈ -4.392305 - 2.359919i; multiplicity 1; Jordan blocks 1 λ ≈ -4.392305 + 2.359919i; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 3; Jordan blocks 1, 2 λ ≈ -3; multiplicity 1; Jordan blocks 1 3
Sector dimension
8
Spectral radius
$25.1996438$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$25.19964$11
$7.584966$11
$-4.392305 - 2.359919\mathrm{i}$11
$-4.392305 + 2.359919\mathrm{i}$11
$-4$31, 2
$-3$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda + 4\right)^{3} \left(\lambda^{4} - 24 \lambda^{3} - 72 \lambda^{2} + 864 \lambda + 4752\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda + 4\right)^{2} \left(\lambda^{4} - 24 \lambda^{3} - 72 \lambda^{2} + 864 \lambda + 4752\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 6 & 42 & 108 & 180 & 528 & 1392 & 3600 \\ 1 & -1 & 3 & 36 & 24 & 144 & 252 & 864 \\ 1 & 1 & 3 & 12 & 28 & 72 & 216 & 528 \\ 0 & 1 & 1 & 2 & 2 & 24 & 36 & 144 \\ 0 & 0 & 3 & 3 & 3 & 12 & 48 & 108 \\ 0 & 0 & 0 & 3 & 1 & 2 & 6 & 36 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 6 \\ 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0\end{pmatrix}\notag\end{equation}$$