Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-141-35-b

·

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

Defining data
Ambient
Fl(1, 2, 6)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
E^(w1) + 4 copies of E^(w2)
Bundle
O(1,0) + 4 copies of O(0,1)
ZL62 label
52.11110
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,5) \\ \det\mathcal{E} &\cong \mathcal{O}(1,4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 35
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 141
$\chi(X, \mathrm{T}_X)$ = -14

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 141
$\mathrm{c}_1^2\mathrm{c}_2$ = 126
$\mathrm{c}_2^2$ = 114
$\mathrm{c}_1\mathrm{c}_3$ = 34
$\mathrm{c}_4$ = 19
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 13 0 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 + 14 0 x0 + 14 x1
$\mathrm{HH}^{ 3 }$ = x1 + 31 0 0 x1 + 11 20
$\mathrm{HH}^{ 4 }$ = 35 0 0 0 0 35
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 30t + 79t^{2} + 30t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(439)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) No match

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( t + 1 \right) ^{2} \left( 16033t^{6} + 18744t^{5} + 8613t^{4} + 1887t^{3} + 114t^{2} - 3t - 1 \right) \left( 48931277964t^{13} - 119850907088t^{12} - 1260281046606t^{11} - 2866616092897t^{10} - 2945699623241t^{9} - 1509218552481t^{8} - 331469810536t^{7} + 13880778935t^{6} + 21039467718t^{5} + 3960419989t^{4} + 371806790t^{3} + 29772417t^{2} + 157180t + 1974 \right)D^{6} + \left( t + 1 \right) \left( 32949637543066104t^{21} - 19631201663035516t^{20} - 1052295219473013248t^{19} - 4204107623626517090t^{18} - 8362177641457095260t^{17} - 10159546275701232845t^{16} - 8101984726001635061t^{15} - 4321689101730332765t^{14} - 1490023495994057918t^{13} - 275475363337420037t^{12} + 9464225687760829t^{11} + 21629009699725366t^{10} + 6644332641199897t^{9} + 1172857213199566t^{8} + 140865755000584t^{7} + 12662703828112t^{6} + 861524432476t^{5} + 39807000310t^{4} + 1862483170t^{3} + 118698320t^{2} + 439956t + 3948 \right)D^{5} + \left( 274580312858884200t^{22} - 68151606145596940t^{21} - 9523106819541629100t^{20} - 42583797094013372982t^{19} - 94320336102418821521t^{18} - 128113467915505856591t^{17} - 114947210024948858915t^{16} - 69661573437405283061t^{15} - 27844546519892175827t^{14} - 6415191577195375496t^{13} - 183368586711101837t^{12} + 437945139959422201t^{11} + 172631788375505695t^{10} + 37236565016843605t^{9} + 5432274461447563t^{8} + 577327853796538t^{7} + 43456600555723t^{6} + 1798558052791t^{5} + 8012234674t^{4} - 1196701459t^{3} - 113464745t^{2} - 480470t - 1974 \right)D^{4} + t \left( 1153237314007313640t^{21} - 877394032792818220t^{20} - 41637021006622065140t^{19} - 173400066328140325770t^{18} - 360408050114472601694t^{17} - 456356918641069761691t^{16} - 376896455267514125280t^{15} - 205585831334809865198t^{14} - 70282880423196176785t^{13} - 11158350014155923819t^{12} + 1956408511101088480t^{11} + 1669929988431410525t^{10} + 495376050112779627t^{9} + 91608098073506773t^{8} + 12030891790116086t^{7} + 1181340942033552t^{6} + 81355979592046t^{5} + 3072161371448t^{4} + 42902815974t^{3} + 622256486t^{2} + 6067924t + 2268 \right)D^{3} + 2 t ^{2} \left( 1274052651665222688t^{20} - 1468980162129126392t^{19} - 47194013547146980216t^{18} - 186148405393142562289t^{17} - 367461363208892298198t^{16} - 438799174128072611224t^{15} - 337529356967382118099t^{14} - 167528356253507367765t^{13} - 48955583405088164602t^{12} - 4144646748557651074t^{11} + 2991478347033369915t^{10} + 1521161482491971351t^{9} + 383652599658473183t^{8} + 64129878000688371t^{7} + 7876988601250292t^{6} + 726878002501385t^{5} + 44834364551055t^{4} + 1393000598357t^{3} + 18302785448t^{2} + 352015256t + 2265024 \right)D^{2} + 4 t ^{2} \left( 691942388404388184t^{20} - 996675170683552012t^{19} - 26054107026935790004t^{18} - 98731959324698353313t^{17} - 187277104109636673366t^{16} - 213425172544108593573t^{15} - 154847088823823240787t^{14} - 70823814185276118389t^{13} - 17712061158554087929t^{12} - 99113941346939417t^{11} + 1687656937633798989t^{10} + 681382764798304038t^{9} + 154330443435979524t^{8} + 24041892490972852t^{7} + 2815830698725706t^{6} + 245467018198255t^{5} + 13541788977813t^{4} + 351372641205t^{3} + 4917958296t^{2} + 92111804t + 426384 \right)D + 8 t ^{2} \left( 141212732327426160t^{20} - 230822265212845080t^{19} - 5371036927642121640t^{18} - 19806157956727153686t^{17} - 36521734198610827516t^{16} - 40223778449570611798t^{15} - 27920028604406200866t^{14} - 11959505493651801951t^{13} - 2585918485596362598t^{12} + 196956343607072118t^{11} + 336081160433786435t^{10} + 118936621733742395t^{9} + 25026432182965794t^{8} + 3711234969895617t^{7} + 419789309285847t^{6} + 34779835653495t^{5} + 1730604332984t^{4} + 38598591506t^{3} + 614599026t^{2} + 9684822t + 31584 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 6)

|λ| = 15.29291 λ ≈ 15.29291; multiplicity 1; Jordan blocks 1 λ ≈ -6.275217 - 6.332648i; multiplicity 1; Jordan blocks 1 λ ≈ -6.275217 + 6.332648i; multiplicity 1; Jordan blocks 1 λ ≈ -1.908651 - 1.791268i; multiplicity 1; Jordan blocks 1 λ ≈ -1.908651 + 1.791268i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ -1.925179; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 2; Jordan blocks 2 2
Sector dimension
9
Spectral radius
$15.2929144$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$15.29291$11
$-6.275217 - 6.332648\mathrm{i}$11
$-6.275217 + 6.332648\mathrm{i}$11
$-1.908651 - 1.791268\mathrm{i}$11
$-1.908651 + 1.791268\mathrm{i}$11
$-2$11
$-1.925179$11
$-1$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda + 2\right) \left(\lambda^{6} + 3 \lambda^{5} - 114 \lambda^{4} - 1887 \lambda^{3} - 8613 \lambda^{2} - 18744 \lambda - 16033\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda + 2\right) \left(\lambda^{6} + 3 \lambda^{5} - 114 \lambda^{4} - 1887 \lambda^{3} - 8613 \lambda^{2} - 18744 \lambda - 16033\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 8 & 33 & 33 & 63 & 96 & 120 & 180 \\ 1 & -1 & 4 & 0 & 0 & 0 & 30 & 30 & 36 \\ 1 & 1 & -1 & 6 & 12 & 12 & 33 & 45 & 36 \\ 0 & 1 & 2 & -1 & 0 & 0 & 0 & 0 & 9 \\ 0 & 2 & 1 & 3 & -1 & 6 & 0 & 0 & 18 \\ 0 & 0 & 1 & 0 & 1 & -1 & 10 & 10 & 9 \\ 0 & 0 & 0 & 1 & 4 & \frac{3}{2} & \frac{5}{2} & \frac{15}{2} & 0 \\ 0 & 0 & 0 & 1 & -\frac{7}{5} & \frac{21}{10} & -\frac{11}{10} & -\frac{9}{2} & \frac{24}{5} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{8}{3} & \frac{10}{3} & 0\end{pmatrix}\notag\end{equation}$$