Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-128-33-d

·

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

Defining data
Ambient
ℙ4P4 × ℙ10P10
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_10node 11: fundamental weight ω_11node 12: fundamental weight ω_12node 13: fundamental weight ω_13node 14: fundamental weight ω_14ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10ω11ω12ω13ω14
Bundle
E^(2w5) + 2 copies of E^(w4 + w5) + E^(2w1)
Bundle
O_2(2) + 2 copies of Q_1 box O_2(1) + O_1(2)
ZL62 label
40A00.Iw0aw0aE
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;11) \\ \det\mathcal{E} &\cong \mathcal{O}(4;10) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 33
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 128
$\chi(X, \mathrm{T}_X)$ = -28

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 128
$\mathrm{c}_1^2\mathrm{c}_2$ = 128
$\mathrm{c}_2^2$ = 120
$\mathrm{c}_1\mathrm{c}_3$ = 16
$\mathrm{c}_4$ = 40
Hodge diamond
1
00
0 2 0
0 0 0 0
0 2 30 2 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 + x1 + 28 0 x0 + 28 x1
$\mathrm{HH}^{ 3 }$ = x1 + 26 0 0 x1 + 6 20
$\mathrm{HH}^{ 4 }$ = 37 0 0 4 0 33
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 28t + 70t^{2} + 28t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(494)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(544)

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( 2t - 1 \right) \left( 2t + 1 \right) ^{3} \left( 1264t^{4} - 1776t^{3} - 128t^{2} - 4t + 1 \right) \left( 128372736t^{9} - 333707264t^{8} - 341984000t^{7} - 130119680t^{6} - 15724864t^{5} - 2530944t^{4} + 256032t^{3} - 29320t^{2} - 531t - 2 \right)D^{6} + \left( 2t + 1 \right) ^{2} \left( 13630103617536t^{15} - 55218455183360t^{14} + 11634150621184t^{13} + 33199309963264t^{12} + 13211936604160t^{11} - 1396430598144t^{10} - 788189556736t^{9} - 239853018112t^{8} - 27346566208t^{7} + 2150641216t^{6} - 392727984t^{5} - 48031456t^{4} + 1308804t^{3} - 112684t^{2} - 1577t - 4 \right)D^{5} + \left( 2t + 1 \right) \left( 227168393625600t^{16} - 815197982556160t^{15} - 263816078721024t^{14} + 406486315352064t^{13} + 343676839837696t^{12} + 76046744698880t^{11} + 3500811919360t^{10} - 1430676284416t^{9} - 812898749824t^{8} + 8207995584t^{7} + 5817127264t^{6} - 1638738928t^{5} - 32246936t^{4} - 2335844t^{3} + 108502t^{2} + 1285t + 2 \right)D^{4} + t \left( 1908214506455040t^{16} - 5949488113909760t^{15} - 5814994487541760t^{14} + 803547794374656t^{13} + 3102427363491840t^{12} + 1497357245448192t^{11} + 314414975991808t^{10} + 36530437234688t^{9} - 3376896080640t^{8} - 1291665451520t^{7} + 89858897152t^{6} - 16184233984t^{5} - 3711220960t^{4} - 98519200t^{3} - 1275552t^{2} - 5744t - 3 \right)D^{3} + 8 t ^{2} \left( 527030673211392t^{15} - 1655010985836544t^{14} - 1668346247364608t^{13} - 71427564478464t^{12} + 593657543938048t^{11} + 301966469648384t^{10} + 70233685284352t^{9} + 10002414490112t^{8} - 557271475968t^{7} - 159683472384t^{6} + 16510366768t^{5} - 4605240064t^{4} - 475624882t^{3} - 10795880t^{2} - 142827t - 660 \right)D^{2} + 16 t ^{2} \left( 286232175968256t^{15} - 903546120372224t^{14} - 937859879206912t^{13} - 147257794707456t^{12} + 232733837223936t^{11} + 124992719582208t^{10} + 31279848269056t^{9} + 4637594767616t^{8} - 284414333056t^{7} - 40335786240t^{6} + 4817562528t^{5} - 2039412776t^{4} - 129326015t^{3} - 2848508t^{2} - 32567t - 138 \right)D + 16 t ^{2} \left( 116829459578880t^{15} - 370091466424320t^{14} - 392735390564352t^{13} - 88475840299008t^{12} + 72877867470848t^{11} + 41334008031232t^{10} + 10979678713344t^{9} + 1620076327424t^{8} - 116294348416t^{7} - 7205071872t^{6} + 862717344t^{5} - 660544720t^{4} - 30137946t^{3} - 675104t^{2} - 6143t - 20 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ4P4 × ℙ10P10

|λ| = 17.19842 λ ≈ 17.19842; multiplicity 1; Jordan blocks 1 λ ≈ -6.938209 - 7.762821i; multiplicity 1; Jordan blocks 1 λ ≈ -6.938209 + 7.762821i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 3; Jordan blocks 3 λ ≈ 2; multiplicity 1; Jordan blocks 1 λ ≈ 0.6779985; multiplicity 1; Jordan blocks 1 3
Sector dimension
8
Spectral radius
$17.198419$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$17.19842$11
$-6.938209 - 7.762821\mathrm{i}$11
$-6.938209 + 7.762821\mathrm{i}$11
$-2$33
$2$11
$0.6779985$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda - 2\right) \left(\lambda^{4} - 4 \lambda^{3} - 128 \lambda^{2} - 1776 \lambda + 1264\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda - 2\right) \left(\lambda^{4} - 4 \lambda^{3} - 128 \lambda^{2} - 1776 \lambda + 1264\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 12 & 96 & 48 & 288 & 160 & 960 \\ 1 & 0 & 4 & -4 & 0 & 144 & 96 & 288 \\ 1 & 0 & 0 & 24 & 16 & 96 & 48 & 96 \\ 0 & 1 & 3 & 0 & 0 & 48 & 32 & 144 \\ 0 & 1 & -1 & 4 & 0 & -32 & -20 & 0 \\ 0 & 0 & 0 & 1 & -1 & -8 & -4 & 12 \\ 0 & 0 & 0 & 1 & 3 & 16 & 8 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{5}{3} & 1 & 0\end{pmatrix}\notag\end{equation}$$