Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-65-20-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ4P4
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_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w2 + 2w3) + E^(w1 + 2w3)
Bundle
O_2(1) box O_3(2) + O_1(1) box O_3(2)
ZL62 label
1140.x351x311
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 20
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 65
$\chi(X, \mathrm{T}_X)$ = -28

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 65
$\mathrm{c}_1^2\mathrm{c}_2$ = 98
$\mathrm{c}_2^2$ = 140
$\mathrm{c}_1\mathrm{c}_3$ = 10
$\mathrm{c}_4$ = 37
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 29 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 + 28 0 x0 + 28 0
$\mathrm{HH}^{ 3 }$ = x1 0 0 x1 0
$\mathrm{HH}^{ 4 }$ = x1 + 8 0 0 x1 - 12 0 20
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 15t + 33t^{2} + 15t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(688)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(686)

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( 6t + 1 \right) \left( 7t + 1 \right) \left( 11t - 1 \right) \left( 5t + 1 \right) ^{2} \left( 8387t^{4} + 3923t^{3} + 571t^{2} + 19t - 1 \right) \left( 371527735510747376t^{13} + 633362341466763488t^{12} + 496028670494775680t^{11} + 234644097252804616t^{10} + 74141857008409218t^{9} + 16336854945217144t^{8} + 2542201443636324t^{7} + 276113194522441t^{6} + 20048416419830t^{5} + 871563370279t^{4} + 15092447064t^{3} - 310874649t^{2} - 12888708t + 19305 \right)D^{6} + 2 \left( 5t + 1 \right) \left( 75578655620508120572128560t^{21} + 186329051196498507754929736t^{20} + 215707549408032652012406400t^{19} + 155327466198805732163237640t^{18} + 77661467493267216689668274t^{17} + 28490807777299679429295035t^{16} + 7897129361568363226431568t^{15} + 1676190704802887650885345t^{14} + 272596133156648159687430t^{13} + 33494018376612185586945t^{12} + 2990713460707531082932t^{11} + 174481288162580430983t^{10} + 3945346788353574004t^{9} - 333832387133882810t^{8} - 39460808533041002t^{7} - 2021300899766023t^{6} - 62515841112816t^{5} - 1315751996025t^{4} - 13860468090t^{3} + 520870191t^{2} + 19294452t - 19305 \right)D^{5} + \left( 6298221301709010047677380000t^{22} + 16153476481068474293963795600t^{21} + 19531239181857358408335530064t^{20} + 14759213083331868504078809696t^{19} + 7790056774164755779130314708t^{18} + 3039781668056105058774707246t^{17} + 905217203713887065547644466t^{16} + 209295715759692407723427448t^{15} + 37838214897053337951198161t^{14} + 5339192431296911802673305t^{13} + 581173814601080890289324t^{12} + 47543218302886450045052t^{11} + 2770755754453384273120t^{10} + 101396957171034806305t^{9} + 1416047741498725107t^{8} - 32339035816808372t^{7} + 1070726611332493t^{6} + 268928600053447t^{5} + 12263348904230t^{4} + 210244959696t^{3} + 43357086t^{2} - 23646639t + 19305 \right)D^{4} + 2 t \left( 13226264733588921100122498000t^{21} + 32876879234924260470327859400t^{20} + 38539443645885830624707662680t^{19} + 28234952767322496045357085112t^{18} + 14443497825757322712716264790t^{17} + 5459177518628850085668790251t^{16} + 1573460440735530211564231761t^{15} + 351816188063341128799916943t^{14} + 61463348313045489048149179t^{13} + 8377937202286080932927789t^{12} + 881438442208522176205124t^{11} + 69877610004282823511576t^{10} + 3975267541801373672706t^{9} + 144667384373837210671t^{8} + 2099136349167026905t^{7} - 69194921571909517t^{6} - 3487281038555801t^{5} - 17762029025149t^{4} + 1358364140538t^{3} + 6782461830t^{2} - 365203722t - 64746 \right)D^{3} + 4 t ^{2} \left( 14611873419964903310611521600t^{20} + 35437493754205766045353588640t^{19} + 40534268434246498372602217424t^{18} + 28968879682351394062905965496t^{17} + 14446915891363626639833488376t^{16} + 5318314143766500139001538564t^{15} + 1491062749340198771288798874t^{14} + 323791844050387229422457973t^{13} + 54834822296532138411046208t^{12} + 7229253900269413701597901t^{11} + 733652876141509666625934t^{10} + 55909277403613521016889t^{9} + 3041893220037782429386t^{8} + 104680199880999904429t^{7} + 1326601730922639012t^{6} - 55543460048580762t^{5} - 2704348870933502t^{4} - 32360800230490t^{3} + 133352343156t^{2} - 3286271952t - 98260500 \right)D^{2} + 8 t ^{2} \left( 7935758840153352660073498800t^{20} + 18894542356041289369813445240t^{19} + 21215213583707154687813545496t^{18} + 14877535844680769599659457888t^{17} + 7274992421378610630963675314t^{16} + 2623218800012994556782352093t^{15} + 719384600942474227346855561t^{14} + 152538378420742118124610322t^{13} + 25169186387499714177211782t^{12} + 3224002117453251943889373t^{11} + 316706836401958630703216t^{10} + 23232974090290993573073t^{9} + 1204665900724006667284t^{8} + 38454341833258726508t^{7} + 355129143697354467t^{6} - 25628826543783547t^{5} - 1083432339389248t^{4} - 14157084405426t^{3} - 42429449844t^{2} - 1888387956t - 13719420 \right)D + 96 t ^{2} \left( 269923770073243287757602000t^{20} + 634590020031401806433078600t^{19} + 703437697750792872889951176t^{18} + 486790543896039837837119728t^{17} + 234735133240079600202224430t^{16} + 83387403255444517361221939t^{15} + 22501259664385961427470779t^{14} + 4687218765298163782990780t^{13} + 758253674149727035168544t^{12} + 94971171118564284414054t^{11} + 9088400458474427977721t^{10} + 645708112922825581111t^{9} + 32062142586726095284t^{8} + 947378378055807849t^{7} + 4946192374683585t^{6} - 795184278152900t^{5} - 29906071459268t^{4} - 402025627842t^{3} - 2376267069t^{2} - 53883687t - 167310 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ4P4

|λ| = 37.2934 λ ≈ 37.2934; multiplicity 1; Jordan blocks 1 λ ≈ 11; multiplicity 2; Jordan blocks 1, 1 λ ≈ -7; multiplicity 1; Jordan blocks 1 λ ≈ -6.824364; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 1; Jordan blocks 1 λ ≈ -5.734519 - 0.2638502i; multiplicity 1; Jordan blocks 1 λ ≈ -5.734519 + 0.2638502i; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 4; Jordan blocks 2, 2 2 4
Sector dimension
12
Spectral radius
$37.2934025$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$37.2934$11
$11$21, 1
$-7$11
$-6.824364$11
$-6$11
$-5.734519 - 0.2638502\mathrm{i}$11
$-5.734519 + 0.2638502\mathrm{i}$11
$-5$42, 2

Characteristic polynomial

$$\begin{equation}\left(\lambda + 5\right)^{4} \left(\lambda + 6\right) \left(\lambda + 7\right) \left(\lambda - 11\right)^{2} \left(\lambda^{4} - 19 \lambda^{3} - 571 \lambda^{2} - 3923 \lambda - 8387\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 5\right)^{2} \left(\lambda + 6\right) \left(\lambda + 7\right) \left(\lambda - 11\right) \left(\lambda^{4} - 19 \lambda^{3} - 571 \lambda^{2} - 3923 \lambda - 8387\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 16 & 72 & 48 & 300 & 300 & 732 & 1344 & 3120 & 3120 & 14100 \\ 1 & -1 & 0 & 6 & 16 & 64 & 24 & 88 & 276 & 582 & 390 & 2448 \\ 1 & 0 & -1 & 6 & 16 & 24 & 64 & 88 & 276 & 390 & 582 & 2448 \\ 1 & 2 & 2 & 4 & 0 & 32 & 32 & 88 & 96 & 324 & 324 & 1344 \\ 0 & 1 & 1 & 0 & -2 & 6 & 6 & 9 & 56 & 76 & 76 & 444 \\ 0 & 1 & 0 & 1 & 2 & 3 & 0 & 8 & 32 & 72 & 32 & 276 \\ 0 & 0 & 1 & 1 & 2 & 0 & 3 & 8 & 32 & 32 & 72 & 276 \\ 0 & 0 & 0 & 1 & 0 & 2 & 2 & 0 & 0 & 16 & 16 & 48 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & -\frac{1}{4} & 2 & \frac{15}{2} & \frac{15}{2} & 56 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{3}{2} & 2 & 0 & 1 & 16 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & 2 & 1 & 0 & 16 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$