Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-256-57-b

·

No geometric description is recorded for this family (yet).

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ5P5
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 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^(2w3) + E^(w2 + w3) + E^(w1 + w3)
Bundle
O_3(2) + O_2(1) box O_3(1) + O_1(1) box O_3(1)
ZL62 label
1150.9w6w1
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;6) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 57
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 256
$\chi(X, \mathrm{T}_X)$ = -1

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 256
$\mathrm{c}_1^2\mathrm{c}_2$ = 160
$\mathrm{c}_2^2$ = 100
$\mathrm{c}_1\mathrm{c}_3$ = 52
$\mathrm{c}_4$ = 16
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 8 0 0
0 0 0 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 + x1 + 1 0 x0 + 1 x1
$\mathrm{HH}^{ 3 }$ = x1 + 43 0 0 x1 - 16 59
$\mathrm{HH}^{ 4 }$ = 57 0 0 0 0 57
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 52t + 150t^{2} + 52t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(223)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(301)

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( t + 1 \right) \left( 2t + 1 \right) \left( 3t - 1 \right) \left( 5t + 1 \right) \left( 16t^{2} + 8t - 1 \right) \left( 48t^{2} + 8t + 1 \right) \left( 62058700800t^{13} + 95915335680t^{12} - 33014062080t^{11} - 178831904256t^{10} - 171984003872t^{9} - 80154263968t^{8} - 18837331524t^{7} - 1524758292t^{6} + 236577668t^{5} + 63509740t^{4} + 3635513t^{3} + 221273t^{2} + 818t + 48 \right)D^{6} + \left( 24307151929344000t^{21} + 77573246209228800t^{20} + 63719408806133760t^{19} - 79065970311168000t^{18} - 226770476817973248t^{17} - 236957318811844608t^{16} - 139719865544972288t^{15} - 47674052629984256t^{14} - 7177771807051840t^{13} + 987370335851296t^{12} + 673318534045160t^{11} + 118983643232524t^{10} + 4313434954524t^{9} - 1381801994240t^{8} - 201954187082t^{7} - 22727295667t^{6} - 4011051861t^{5} - 441109747t^{4} - 18679315t^{3} - 892234t^{2} - 1974t - 96 \right)D^{5} + \left( 164430733639680000t^{21} + 476161704670003200t^{20} + 290321936288317440t^{19} - 668743692018647040t^{18} - 1529358985243754496t^{17} - 1494002964675600384t^{16} - 856382039594749952t^{15} - 298615534046037504t^{14} - 56164619847953216t^{13} - 1382493322437472t^{12} + 1796598003306904t^{11} + 330028908234188t^{10} - 7117075892688t^{9} - 9606676778900t^{8} - 1402222995966t^{7} - 113381209135t^{6} - 3036228542t^{5} + 373715768t^{4} + 16890844t^{3} + 906381t^{2} + 3622t + 48 \right)D^{4} + t \left( 564783824240640000t^{20} + 1510381415301120000t^{19} + 645659976652554240t^{18} - 2611017658485964800t^{17} - 5201673183098044416t^{16} - 4809057471409373184t^{15} - 2660771560594884608t^{14} - 909980039508772864t^{13} - 174444581750002880t^{12} - 8077243133610912t^{11} + 4276618123173784t^{10} + 965178974848948t^{9} + 51942833966164t^{8} - 6104745477520t^{7} - 1086666875918t^{6} - 123185021157t^{5} - 6584583903t^{4} - 129672693t^{3} - 3630275t^{2} - 54638t - 18 \right)D^{3} + 2 t ^{2} \left( 517599352848384000t^{19} + 1299778106897203200t^{18} + 358304079052800000t^{17} - 2590887623877918720t^{16} - 4711287011137634304t^{15} - 4166316374474256384t^{14} - 2225042854809054208t^{13} - 735432689207282944t^{12} - 134910540927255104t^{11} - 4619530589810112t^{10} + 3842331100924920t^{9} + 884954135792540t^{8} + 75550747510996t^{7} + 1730894496716t^{6} - 272482841366t^{5} - 62102674715t^{4} - 3040982845t^{3} - 63435880t^{2} - 2144479t - 21264 \right)D^{2} + 16 t ^{2} \left( 59695505473536000t^{19} + 142877389927219200t^{18} + 21983905777582080t^{17} - 314754149599395840t^{16} - 537775062858067968t^{15} - 459332168054430720t^{14} - 237706490019918848t^{13} - 75715198798828608t^{12} - 12977111417512032t^{11} - 117011550790992t^{10} + 474781203980528t^{9} + 104565345380080t^{8} + 10290584871924t^{7} + 572450970268t^{6} + 422460660t^{5} - 4546583372t^{4} - 196771833t^{3} - 4841638t^{2} - 158873t - 1104 \right)D + 64 t ^{2} \left( 5361871749120000t^{19} + 12393090102067200t^{18} + 761922588672000t^{17} - 29260169703751680t^{16} - 47930872124436480t^{15} - 39898485892998144t^{14} - 20128488205680000t^{13} - 6200434593689088t^{12} - 987693414191152t^{11} + 21005001636488t^{10} + 45580093529848t^{9} + 9549725310424t^{8} + 986942513824t^{7} + 66140236014t^{6} + 1135002792t^{5} - 292260444t^{4} - 11247001t^{3} - 339864t^{2} - 9729t - 48 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ5P5

|λ| = 9.656854 λ ≈ 9.656854; multiplicity 1; Jordan blocks 1 λ ≈ -4 - 5.656854i; multiplicity 1; Jordan blocks 1 λ ≈ -4 + 5.656854i; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 2; Jordan blocks 1, 1 λ ≈ 3; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ -1.656854; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 2; Jordan blocks 1, 1 λ ≈ 0; multiplicity 1; Jordan blocks 1 2 2 2
Sector dimension
12
Spectral radius
$9.65685425$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$9.656854$11
$-4 - 5.656854\mathrm{i}$11
$-4 + 5.656854\mathrm{i}$11
$-5$21, 1
$3$21, 1
$-2$11
$-1.656854$11
$-1$21, 1
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right)^{2} \left(\lambda + 2\right) \left(\lambda + 5\right)^{2} \left(\lambda - 3\right)^{2} \left(\lambda^{2} + 8 \lambda + 48\right) \left(\lambda^{2} - 8 \lambda - 16\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right) \left(\lambda + 2\right) \left(\lambda + 5\right) \left(\lambda - 3\right) \left(\lambda^{2} + 8 \lambda + 48\right) \left(\lambda^{2} - 8 \lambda - 16\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 4 & 0 & 6 & 6 & 12 & 8 & 8 & 8 & 0 \\ 1 & -1 & 0 & 0 & 0 & 4 & 0 & 4 & 6 & 6 & 6 & 0 \\ 1 & 0 & -1 & 0 & 0 & 0 & 4 & 4 & 6 & 6 & 6 & 0 \\ 2 & 1 & 1 & 0 & 0 & 0 & 0 & 4 & 0 & 6 & 6 & 8 \\ 0 & 1 & 1 & 0 & -2 & 0 & 0 & 0 & 4 & 4 & 4 & 0 \\ 0 & 2 & 0 & 1 & 1 & -1 & 0 & 0 & 0 & 4 & 0 & 6 \\ 0 & 0 & 2 & 1 & 1 & 0 & -1 & 0 & 0 & 0 & 4 & 6 \\ 0 & 0 & 0 & 2 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 1 & 1 & -2 & -2 & -1 & -1 & 4 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 3 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 3 & 0\end{pmatrix}\notag\end{equation}$$