Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 4-84-24-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ2P2
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_4 (marked)node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
E^(w1 + w2 + w3 + 2w4)
Bundle
O_1(1) box O_2(1) box O_3(1) box O_4(2)
ZL62 label
11120.x5001
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;2;3) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;1;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 24
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 84
$\chi(X, \mathrm{T}_X)$ = -30

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 84
$\mathrm{c}_1^2\mathrm{c}_2$ = 108
$\mathrm{c}_2^2$ = 138
$\mathrm{c}_1\mathrm{c}_3$ = 16
$\mathrm{c}_4$ = 58
Hodge diamond
1
00
0 4 0
0 0 0 0
0 3 42 3 0
0 0 0 0
0 4 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = 30 0 30 0
$\mathrm{HH}^{ 3 }$ = 3 0 0 3 0
$\mathrm{HH}^{ 4 }$ = 30 0 0 6 0 24
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 19t + 44t^{2} + 19t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(660)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(667)

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( 5t + 1 \right) ^{4} \left( 31t^{2} + t - 1 \right) \left( 1003t^{3} + 196t^{2} + 20t - 1 \right) \left( 1041817051536t^{10} + 1389741223452t^{9} + 841695366550t^{8} + 306642219642t^{7} + 73604696192t^{6} + 11713686658t^{5} + 1164090498t^{4} + 61727084t^{3} + 980737t^{2} - 23934t + 77 \right)D^{6} + \left( 5t + 1 \right) ^{3} \left( 3401287846257929040t^{16} + 5639257312519488984t^{15} + 4300801764686516590t^{14} + 1993177519196582990t^{13} + 617641930288502200t^{12} + 130398773351561048t^{11} + 17695857334795616t^{10} + 1136127978609320t^{9} - 72954067442945t^{8} - 23943149345735t^{7} - 2399174794898t^{6} - 128786304066t^{5} - 4877800486t^{4} - 206205780t^{3} - 4833943t^{2} + 72803t - 154 \right)D^{5} + \left( 5t + 1 \right) ^{2} \left( 141720326927413710000t^{17} + 251561247825900892200t^{16} + 207335618576923577758t^{15} + 105243510476388922892t^{14} + 36500627556346937770t^{13} + 8981010561059688994t^{12} + 1560662698889628528t^{11} + 181923643492088094t^{10} + 11989001835822085t^{9} + 64941360791160t^{8} - 59497204818834t^{7} - 4362350203696t^{6} - 51558178982t^{5} + 7987956779t^{4} + 409238708t^{3} + 5455620t^{2} - 40113t + 77 \right)D^{4} + t \left( 5t + 1 \right) \left( 2976126865475687910000t^{17} + 5684105685348807120000t^{16} + 5071805112273181438450t^{15} + 2808973298830477773622t^{14} + 1075284474319699884522t^{13} + 297661720821791640726t^{12} + 60239016688189584860t^{11} + 8778102463529873588t^{10} + 878857362183916125t^{9} + 53551794430391967t^{8} + 1130216119242693t^{7} - 85464447265297t^{6} - 6571715630233t^{5} - 112942750428t^{4} + 2013184617t^{3} + 17329825t^{2} - 1399346t - 363 \right)D^{3} + t ^{2} \left( 32879115847159980720000t^{17} + 67732690938713139913500t^{16} + 65468159818623434264800t^{15} + 39478569757243464439400t^{14} + 16576341687965892823368t^{13} + 5093258044207873007148t^{12} + 1166568678308996484424t^{11} + 198755543012323458220t^{10} + 24688760805759093124t^{9} + 2138495868085313913t^{8} + 116876646256188819t^{7} + 2854989461237258t^{6} - 62746329282914t^{5} - 5566984935807t^{4} - 75667394553t^{3} + 479993828t^{2} - 48725548t - 844228 \right)D^{2} + 2 t ^{2} \left( 17856761192854127460000t^{17} + 36133553118127595044500t^{16} + 34332344731725829276300t^{15} + 20373382436006133075580t^{14} + 8423912877597067656360t^{13} + 2547790484766742466624t^{12} + 573400956072450301236t^{11} + 95738523126672199564t^{10} + 11626837513046787402t^{9} + 985583848938799155t^{8} + 53535411766710477t^{7} + 1459361533765745t^{6} - 6750098147137t^{5} - 1394850707304t^{4} - 18292195424t^{3} - 100863180t^{2} - 19591666t - 133056 \right)D + 4 t ^{2} \left( 3644236978133495400000t^{17} + 7284256716088076355000t^{16} + 6839491401850638262200t^{15} + 4013257604122519384020t^{14} + 1641254524398437152848t^{13} + 490638952589966909900t^{12} + 108937334775183704200t^{11} + 17891549986473143800t^{10} + 2129829513342547356t^{9} + 176437199113803890t^{8} + 9373977122995958t^{7} + 256613420074343t^{6} - 162474250579t^{5} - 183463992379t^{4} - 2922383637t^{3} - 49281722t^{2} - 2676806t - 9856 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ2P2

|λ| = 28.20869 λ ≈ 28.20869; multiplicity 1; Jordan blocks 1 λ ≈ 6.09017; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ -4.104345 - 4.325594i; multiplicity 1; Jordan blocks 1 λ ≈ -4.104345 + 4.325594i; multiplicity 1; Jordan blocks 1 λ ≈ -5.09017; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ -5; multiplicity 8; Jordan blocks 1, 1, 1, 1, 1, 3 3 3 8
Sector dimension
17
Spectral radius
$28.2086907$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$28.20869$11
$6.09017$31, 1, 1
$-4.104345 - 4.325594\mathrm{i}$11
$-4.104345 + 4.325594\mathrm{i}$11
$-5.09017$31, 1, 1
$-5$81, 1, 1, 1, 1, 3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 5\right)^{8} \left(\lambda^{2} - \lambda - 31\right)^{3} \left(\lambda^{3} - 20 \lambda^{2} - 196 \lambda - 1003\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 5\right)^{3} \left(\lambda^{2} - \lambda - 31\right) \left(\lambda^{3} - 20 \lambda^{2} - 196 \lambda - 1003\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 12 & 12 & 12 & 28 & 42 & 42 & 102 & 42 & 102 & 102 & 99 & 192 & 416 & 416 & 416 & 1800 \\ 1 & -1 & 1 & 1 & 3 & 10 & 10 & 24 & 2 & 12 & 12 & 16 & 36 & 84 & 84 & 54 & 320 \\ 1 & 1 & -1 & 1 & 3 & 10 & 2 & 12 & 10 & 24 & 12 & 16 & 36 & 84 & 54 & 84 & 320 \\ 1 & 1 & 1 & -1 & 3 & 2 & 10 & 12 & 10 & 12 & 24 & 16 & 36 & 54 & 84 & 84 & 320 \\ 1 & 2 & 2 & 2 & 0 & 4 & 4 & 12 & 4 & 12 & 12 & 12 & 12 & 42 & 42 & 42 & 192 \\ 0 & 1 & 1 & 0 & 0 & -2 & 1 & 3 & 1 & 3 & 0 & 2 & 8 & 20 & 12 & 12 & 66 \\ 0 & 1 & 0 & 1 & 0 & 1 & -2 & 3 & 1 & 0 & 3 & 2 & 8 & 12 & 20 & 12 & 66 \\ 0 & 1 & 0 & 0 & 1 & 2 & 2 & -1 & 0 & 1 & 1 & 1 & 4 & 10 & 10 & 2 & 36 \\ 0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & -2 & 3 & 3 & 2 & 8 & 12 & 12 & 20 & 66 \\ 0 & 0 & 1 & 0 & 1 & 2 & 0 & 1 & 2 & -1 & 1 & 1 & 4 & 10 & 2 & 10 & 36 \\ 0 & 0 & 0 & 1 & 1 & 0 & 2 & 1 & 2 & 1 & -1 & 1 & 4 & 2 & 10 & 10 & 36 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 2 & 0 & 2 & 2 & -2 & 0 & 4 & 4 & 4 & 12 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & -\frac{1}{2} & 1 & -\frac{1}{2} & -\frac{1}{2} & -\frac{3}{2} & -3 & 1 & 1 & 1 & 10 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{3}{2} & 0 & \frac{3}{2} & 0 & 1 & 2 & 0 & 2 & 2 & 12 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & 0 & 0 & \frac{3}{2} & 1 & 2 & 2 & 0 & 2 & 12 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & \frac{3}{2} & 1 & 2 & 2 & 2 & 0 & 12 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & \frac{3}{2} & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$