Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-90-25-b

·

This is a Fano fourfold of K3 type, appearing as case R-64 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ4P4 × ℙ6P6
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9node 10: fundamental weight ω_10ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10
Bundle
E^(2w5) + E^(w4 + w5) + E^(3w1)
Bundle
O_2(2) + Q_1 box O_2(1) + O_1(3)
ZL62 label
40600.EwOK
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;7) \\ \det\mathcal{E} &\cong \mathcal{O}(4;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 25
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 90
$\chi(X, \mathrm{T}_X)$ = -23

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 90
$\mathrm{c}_1^2\mathrm{c}_2$ = 108
$\mathrm{c}_2^2$ = 126
$\mathrm{c}_1\mathrm{c}_3$ = 12
$\mathrm{c}_4$ = 12
Hodge diamond
1
00
0 3 0
0 5 5 0
0 1 22 1 0
0 5 5 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x2 + 23 0 x0 + 23 x2
$\mathrm{HH}^{ 3 }$ = x1 + x4 0 0 x1 x4
$\mathrm{HH}^{ 4 }$ = x1 - x2 + x3 + 1 0 0 x1 - x2 - 24 x3 25
$\mathrm{HH}^{ 5 }$ = x3 - x4 + 13 0 0 x3 - x4 + 13 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) CKK(643)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(651)

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( 29t + 1 \right) \left( 2t - 1 \right) ^{2} \left( 2t + 1 \right) ^{3} \left( 428t^{2} + 4t - 1 \right) \left( 16541205950464t^{9} + 9630284765184t^{8} + 2369966530944t^{7} + 221622838080t^{6} + 5936273504t^{5} - 1106492592t^{4} + 84107656t^{3} + 16584636t^{2} + 19490t + 375 \right)D^{6} + 2 \left( 2t - 1 \right) \left( 2t + 1 \right) ^{2} \left( 8622996826800685056t^{14} + 4744076510471553024t^{13} + 914212077450456064t^{12} - 69008372767299584t^{11} - 48927018317839104t^{10} - 6979602468455424t^{9} - 287293734900096t^{8} + 32026496466432t^{7} - 4856122555744t^{6} - 936509977792t^{5} - 15471245220t^{4} - 193651656t^{3} - 33435687t^{2} - 18360t - 375 \right)D^{5} + \left( 2t + 1 \right) \left( 574866455120045670400t^{16} + 281383077641350021120t^{15} - 16204193013544722432t^{14} - 44092182211411881984t^{13} - 12420755412829136896t^{12} - 1075174101399467008t^{11} + 36841740053999104t^{10} + 12791587259595520t^{9} - 2013954540804096t^{8} - 381013372219008t^{7} + 22395155939488t^{6} + 5007187642672t^{5} + 75377666776t^{4} + 1302688252t^{3} - 60328950t^{2} - 85755t - 375 \right)D^{4} + 2 t \left( 2414439111504191815680t^{16} + 2273895805758479400960t^{15} + 633780715292773130240t^{14} - 59662964659932917760t^{13} - 79613588245812899840t^{12} - 18921908949754229760t^{11} - 1858640619576658944t^{10} - 46126836545859840t^{9} + 287788865342976t^{8} - 3134227807414656t^{7} - 252591925662944t^{6} + 26348718099888t^{5} + 3640600993392t^{4} + 18844556556t^{3} + 250128952t^{2} + 1999221t + 250 \right)D^{3} + 4 t ^{2} \left( 2667380351757011910656t^{15} + 2414798937764161191936t^{14} + 762220556268457345024t^{13} + 26163363755136614400t^{12} - 47515470926013993984t^{11} - 12425138019676119552t^{10} - 1248453912778096128t^{9} - 19949331963820800t^{8} + 477605726706560t^{7} - 2255127461544768t^{6} - 185280297847936t^{5} + 10224822895008t^{4} + 1314117825852t^{3} + 3385608246t^{2} + 145947170t + 708765 \right)D^{2} + 16 t ^{2} \left( 724331733451257544704t^{15} + 636377336465554833408t^{14} + 214952806264487907328t^{13} + 20182247809886238720t^{12} - 6880208302277595648t^{11} - 2072612004687113472t^{10} - 200687198214493056t^{9} + 3113730913609920t^{8} + 483431231954976t^{7} - 443949081960048t^{6} - 42976542145864t^{5} + 1135472277852t^{4} + 118849103000t^{3} + 212125440t^{2} + 19010725t + 65025 \right)D + 32 t ^{2} \left( 147822802745154600960t^{15} + 127202873024341278720t^{14} + 44599068354044239872t^{13} + 5510701218215485440t^{12} - 740434052955701248t^{11} - 278711401787836416t^{10} - 25780773408322304t^{9} + 1486256325563520t^{8} + 117354331326144t^{7} - 74116106920608t^{6} - 8137476790288t^{5} + 127484169288t^{4} + 8286193092t^{3} + 27806010t^{2} + 1940950t + 4875 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ4P4 × ℙ6P6

|λ| = 29 λ ≈ -29; multiplicity 1; Jordan blocks 1 λ ≈ 22.78461; multiplicity 1; Jordan blocks 1 λ ≈ -18.78461; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 3; Jordan blocks 3 λ ≈ 2; multiplicity 2; Jordan blocks 2 3 2
Sector dimension
8
Spectral radius
$29$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$-29$11
$22.78461$11
$-18.78461$11
$-2$33
$2$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda + 29\right) \left(\lambda - 2\right)^{2} \left(\lambda^{2} - 4 \lambda - 428\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda + 29\right) \left(\lambda - 2\right)^{2} \left(\lambda^{2} - 4 \lambda - 428\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 24 & 28 & 72 & 0 & 1152 & 576 & 1440 \\ 1 & -6 & 2 & 64 & 168 & 72 & -864 & 576 \\ 1 & 6 & 0 & 24 & -108 & 108 & 972 & 576 \\ 0 & 1 & 2 & 0 & 15 & 60 & -108 & 0 \\ 0 & 1 & 0 & 2 & -15 & 28 & 132 & 72 \\ 0 & 0 & 0 & 2 & 1 & 0 & 6 & 24 \\ 0 & 0 & 0 & 0 & 1 & 2 & -6 & 4 \\ 0 & 0 & 0 & 0 & 0 & 2 & 1 & 0\end{pmatrix}\notag\end{equation}$$