Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 4-290-63-a

·

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

Defining data
Ambient
Fl(1, 2, 4) × ℙ2P2 × ℙ2P2
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6 (marked)node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(w1) + E^(w3 + w6) + E^(w3 + w4)
Bundle
O_1(1,0) + Q_1,1^v ot U_1,2^v ot U_1,3^v box O_3(1) + Q_1,1^v ot U_1,2^v ot U_1,3^v box O_2(1)
ZL62 label
202032.4wEw5
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,3;3;3) \\ \det\mathcal{E} &\cong \mathcal{O}(1,2;2;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 63
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 290
$\chi(X, \mathrm{T}_X)$ = 6

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 290
$\mathrm{c}_1^2\mathrm{c}_2$ = 164
$\mathrm{c}_2^2$ = 104
$\mathrm{c}_1\mathrm{c}_3$ = 60
$\mathrm{c}_4$ = 18
Hodge diamond
1
00
0 4 0
0 0 0 0
0 0 8 0 0
0 0 0 0
0 4 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 - 6 0 x0 - 6 x1
$\mathrm{HH}^{ 3 }$ = x1 + 40 0 0 x1 - 30 70
$\mathrm{HH}^{ 4 }$ = 63 0 0 0 0 63
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 58t + 172t^{2} + 58t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(26)
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( 89t^{4} + 59t^{3} + 17t^{2} + 5t + 1 \right) \left( 6091t^{6} + 3200t^{5} + 649t^{4} + 384t^{3} - 3t^{2} + t - 1 \right) \left( 5816264239564668t^{17} + 21505196385440288t^{16} + 11594067335622108t^{15} - 9837750133402736t^{14} - 21080478585950220t^{13} - 19112074645475066t^{12} - 9232020890852490t^{11} - 1896578340130062t^{10} + 222745611752090t^{9} + 167476573675556t^{8} + 25605941939258t^{7} + 1593592038112t^{6} - 112504223007t^{5} - 72066294109t^{4} - 7382538090t^{3} - 560203250t^{2} - 3505300t - 118900 \right)D^{6} + \left( 66212811588079106120772t^{27} + 320167971039631737614428t^{26} + 420383866729628325913884t^{25} + 127552623289130372495372t^{24} - 316101699995546936946440t^{23} - 582350286936831612014292t^{22} - 545000762307547619039346t^{21} - 333166171120815923403662t^{20} - 140435115315712200966810t^{19} - 40738510281802856404430t^{18} - 7052741143935708430240t^{17} + 187980712126564943448t^{16} + 613533740986216386025t^{15} + 223829089934704726399t^{14} + 46395827533304595720t^{13} + 5717861821824598124t^{12} + 194196124961204545t^{11} - 77797261496814679t^{10} - 16521165234197784t^{9} - 1853935891598395t^{8} - 205865089428459t^{7} - 20976412811828t^{6} - 2638289302071t^{5} - 442773344714t^{4} - 37038595950t^{3} - 2216090800t^{2} - 9564700t - 237800 \right)D^{5} + \left( 551773429900659217673100t^{27} + 2645875205194306241459860t^{26} + 3182565365094419207375324t^{25} + 416430688595045513046172t^{24} - 3315919655994250968914360t^{23} - 5285835199787241399233970t^{22} - 4596340632154969672074710t^{21} - 2568073733782217113534576t^{20} - 927881327842114968970506t^{19} - 194904088689639927775342t^{18} - 5711921500680077453182t^{17} + 13000942236097028071682t^{16} + 5795187307644577912991t^{15} + 1437703684763366554660t^{14} + 222352524880232611912t^{13} + 18144154293983270158t^{12} - 831568542594390714t^{11} - 627104155896864393t^{10} - 149239704193779523t^{9} - 24405903287317959t^{8} - 2612291291765948t^{7} - 189841311467871t^{6} - 7345381580983t^{5} + 395020569907t^{4} + 32560564250t^{3} + 2224450410t^{2} + 11327900t + 118900 \right)D^{4} + t \left( 2317448405582768714227020t^{26} + 11039444835367424026552180t^{25} + 12261754143298411165678580t^{24} - 437301667149639169349116t^{23} - 16135927608180186240417360t^{22} - 23557339720376077507773152t^{21} - 19439741566832452139022090t^{20} - 10104882441805148407448470t^{19} - 3158705426253206411567198t^{18} - 421092607268743491199256t^{17} + 97711281440574134834244t^{16} + 70963042421932077395860t^{15} + 21574367662677081549023t^{14} + 4139658331767599381867t^{13} + 470401607386269238578t^{12} + 8252509689878213374t^{11} - 9746480198665195069t^{10} - 2592907585009447282t^{9} - 415538268295072302t^{8} - 49287854575160487t^{7} - 4166650010558429t^{6} - 262788410670674t^{5} - 10746714353313t^{4} - 151132154068t^{3} - 3823393890t^{2} - 19839120t - 36900 \right)D^{3} + t ^{2} \left( 5120457429478117540006368t^{25} + 24268120901053379389188496t^{24} + 25166434164485684304296032t^{23} - 4739322354852019393297504t^{22} - 39345996805606429454018600t^{21} - 54281796226192562296399188t^{20} - 43139199125651142144697286t^{19} - 21177493714583879547016344t^{18} - 5797373007555661929907870t^{17} - 349275562444388661102240t^{16} + 364681340766603653759016t^{15} + 165608483555023246810728t^{14} + 40390829396542424888436t^{13} + 6320286005394781529538t^{12} + 520839623159430175841t^{11} - 30949146433709391706t^{10} - 21529734666414183805t^{9} - 4765876479444192828t^{8} - 690846008536066108t^{7} - 73840339613195598t^{6} - 5529533558526270t^{5} - 308835227284136t^{4} - 11419502179036t^{3} - 201818839024t^{2} - 4917650980t - 12184640 \right)D^{2} + 2 t ^{2} \left( 2780938086699322457072424t^{25} + 13130829896727128078895656t^{24} + 12885296713763197805088248t^{23} - 4099003529843924670955304t^{22} - 22828056593603871522718040t^{21} - 30354992373701433960123524t^{20} - 23492629112086026782957126t^{19} - 11042232749611601012672804t^{18} - 2694606141439307315838486t^{17} + 17716496177561810596336t^{16} + 243088993362175133191300t^{15} + 89442411880701294021048t^{14} + 18784715556275596984538t^{13} + 2488662987244610311692t^{12} + 139818814248646755707t^{11} - 26817684941131690690t^{10} - 11088280128979824357t^{9} - 2245069484187931456t^{8} - 304720772812640022t^{7} - 30053356073468736t^{6} - 2027420270000962t^{5} - 99862108407040t^{4} - 3345181174970t^{3} - 70082873880t^{2} - 1455200700t - 1426800 \right)D + 24 t ^{3} \left( 94589730840113008743960t^{24} + 445494662126639140166840t^{23} + 420064894757289999554504t^{22} - 174810171215089485311192t^{21} - 809842768677723668329160t^{20} - 1052406324530910170956868t^{19} - 800377653557959260287390t^{18} - 364979128355493202489820t^{17} - 81431570885249802703374t^{16} + 5046230652923633070760t^{15} + 9116858490117463173136t^{14} + 2982802841846980974864t^{13} + 565539991215001640954t^{12} + 65737610998707232464t^{11} + 2268618965511160835t^{10} - 1041562901837346802t^{9} - 359893810470520497t^{8} - 69343980705805576t^{7} - 9004886388592500t^{6} - 837347135751180t^{5} - 51907201090738t^{4} - 2282198153500t^{3} - 70457940470t^{2} - 1668704400t - 28892700 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 4) × ℙ2P2 × ℙ2P2

|λ| = 8.260782 λ ≈ 8.260782; multiplicity 1; Jordan blocks 1 λ ≈ -2.838679 - 6.122473i; multiplicity 1; Jordan blocks 1 λ ≈ -2.838679 + 6.122473i; multiplicity 1; Jordan blocks 1 λ ≈ 0.206542 - 3.383479i; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ 0.206542 + 3.383479i; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ 0.119567 - 2.97806i; multiplicity 1; Jordan blocks 1 λ ≈ 0.119567 + 2.97806i; multiplicity 1; Jordan blocks 1 λ ≈ -2.706542 - 0.6481502i; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ -2.706542 + 0.6481502i; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ -1.822558; multiplicity 1; Jordan blocks 1 3 3 3 3
Sector dimension
18
Spectral radius
$8.26078237$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$8.260782$11
$-2.838679 - 6.122473\mathrm{i}$11
$-2.838679 + 6.122473\mathrm{i}$11
$0.206542 - 3.383479\mathrm{i}$31, 1, 1
$0.206542 + 3.383479\mathrm{i}$31, 1, 1
$0.119567 - 2.97806\mathrm{i}$11
$0.119567 + 2.97806\mathrm{i}$11
$-2.706542 - 0.6481502\mathrm{i}$31, 1, 1
$-2.706542 + 0.6481502\mathrm{i}$31, 1, 1
$-1.822558$11

Characteristic polynomial

$$\begin{equation}\left(\lambda^{4} + 5 \lambda^{3} + 17 \lambda^{2} + 59 \lambda + 89\right)^{3} \left(\lambda^{6} - \lambda^{5} + 3 \lambda^{4} - 384 \lambda^{3} - 649 \lambda^{2} - 3200 \lambda - 6091\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda^{4} + 5 \lambda^{3} + 17 \lambda^{2} + 59 \lambda + 89\right) \left(\lambda^{6} - \lambda^{5} + 3 \lambda^{4} - 384 \lambda^{3} - 649 \lambda^{2} - 3200 \lambda - 6091\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 3 & 3 & 6 & 6 & 6 & 9 & 3 & 4 & 4 & 4 & 4 & 5 \\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 4 & 0 & 0 & 3 & 3 & 3 & 0 \\ 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & -2 & -2 & -2 & -6 & 0 & 0 & 0 & 0 & 0 & 4 \\ 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 4 & 0 & 3 & 0 & 3 & 3 & 0 \\ 1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 2 & 2 & 0 & 4 & 0 & 3 & 3 & 0 & 3 & 0 \\ 0 & 1 & 2 & -1 & -1 & 0 & 1 & 1 & 0 & 1 & 1 & -1 & 0 & -2 & -2 & -2 & -6 & -6 \\ 0 & 1 & 0 & 1 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 2 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & -2 & 0 & 0 & 0 & 1 & 0 & 0 & 2 & 0 & 2 & 0 \\ 0 & 2 & 1 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 1 & 2 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 1 & 0 & 2 & 0 & 0 & 1 & 0 & 0 & -1 & -1 & 1 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -2 & 2 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 3 & 3 & 3 & 1 & 1 & 0 & 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 3 & 1 & 0 & 4 & 1 & 0 & 3 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 3 & 0 & 1 & 4 & 0 & 3 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & -1 & -1 & 0 & 1 & 1 & 4 & -1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 3 & 3 & 4 & 0\end{pmatrix}\notag\end{equation}$$