Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-99-27-a

·

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

Defining data
Ambient
ℙ4P4 × ℙ7P7
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_11ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10ω11
Bundle
3 copies of E^(w1 + w5) + E^(w4 + w5)
Bundle
3 copies of O_1(1) box O_2(1) + Q_1 box O_2(1)
ZL62 label
40700.wRw3w3w3
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;8) \\ \det\mathcal{E} &\cong \mathcal{O}(4;7) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 27
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 99
$\chi(X, \mathrm{T}_X)$ = -33

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 99
$\mathrm{c}_1^2\mathrm{c}_2$ = 114
$\mathrm{c}_2^2$ = 136
$\mathrm{c}_1\mathrm{c}_3$ = 6
$\mathrm{c}_4$ = 51
Hodge diamond
1
00
0 2 0
0 0 0 0
0 3 39 3 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = x0 + 33 0 33 x0
$\mathrm{HH}^{ 3 }$ = x0 + 16 0 0 x0 + 3 13
$\mathrm{HH}^{ 4 }$ = 33 0 0 6 0 27
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 22t + 53t^{2} + 22t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(586)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(611)

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( t + 1 \right) ^{4} \left( 1031t^{5} + 3730t^{4} + 3629t^{3} + 303t^{2} - 3t - 1 \right) \left( 34549838667t^{10} + 133325384763t^{9} + 272931418318t^{8} + 420834307404t^{7} + 459741825028t^{6} + 291308448463t^{5} + 76213449373t^{4} - 5223359883t^{3} - 3201285129t^{2} - 33913620t - 160380 \right)D^{6} + \left( t + 1 \right) ^{3} \left( 748038556979217t^{16} + 5499858716394252t^{15} + 18806056632483130t^{14} + 41924161975603653t^{13} + 69827414735893442t^{12} + 87699442940633108t^{11} + 77955877103624901t^{10} + 44648186995137899t^{9} + 13970133245971946t^{8} + 1146591579678754t^{7} - 500934426944812t^{6} - 116397772281798t^{5} - 4586089068024t^{4} - 54003493374t^{3} - 12648617196t^{2} - 99014400t - 320760 \right)D^{5} + \left( t + 1 \right) ^{2} \left( 6233654641493475t^{17} + 46458499885613655t^{16} + 164413171437947160t^{15} + 385520657515106505t^{14} + 680404910681701603t^{13} + 913863645140880105t^{12} + 889574848536416126t^{11} + 585233354681837469t^{10} + 233647382902954765t^{9} + 41727935630818365t^{8} - 4006309563751585t^{7} - 2917736288948512t^{6} - 392018237133018t^{5} - 11585586180988t^{4} - 161359250280t^{3} + 11862085647t^{2} + 90056880t + 160380 \right)D^{4} + t \left( t + 1 \right) \left( 26181349494272595t^{17} + 202802923618604610t^{16} + 756193728743791685t^{15} + 1880176711415823967t^{14} + 3526804297110185166t^{13} + 5082600132822068753t^{12} + 5434438990497055193t^{11} + 4091088415045182109t^{10} + 2015735095309559025t^{9} + 555379112592151183t^{8} + 33426555123540072t^{7} - 25842974089751202t^{6} - 7213434767172966t^{5} - 651483050651322t^{4} - 13220796573558t^{3} - 174888323226t^{2} - 960693960t - 204120 \right)D^{3} + 2 t ^{2} \left( 28924157536529724t^{17} + 237332710021578402t^{16} + 943621235712771635t^{15} + 2501490182270114205t^{14} + 4999507374160807569t^{13} + 7743080932587581354t^{12} + 9089748105782326088t^{11} + 7777008658385451466t^{10} + 4606372927293503528t^{9} + 1724187348845315629t^{8} + 307887949459973283t^{7} - 28199766044189508t^{6} - 25528348178439874t^{5} - 4666867057744632t^{4} - 305617697041015t^{3} - 5552885623652t^{2} - 77771681154t - 370608480 \right)D^{2} + 4 t ^{2} \left( 15708809696563557t^{17} + 122671373212417533t^{16} + 468925650314663922t^{15} + 1210807634189831229t^{14} + 2374562949746480082t^{13} + 3595906120712782807t^{12} + 4081282783145287280t^{11} + 3326122324967318846t^{10} + 1839482912728459396t^{9} + 618393845197595195t^{8} + 82840261462320282t^{7} - 19559648997388992t^{6} - 9875239468831482t^{5} - 1488867940634862t^{4} - 76600430207673t^{3} - 1407601344792t^{2} - 17899725600t - 70567200 \right)D + 24 t ^{2} \left( 1068626509970310t^{17} + 8058904093064190t^{16} + 29969149361408400t^{15} + 76004252865648348t^{14} + 147103826579988648t^{13} + 219039611293918232t^{12} + 242200709700429349t^{11} + 189992398114608781t^{10} + 99421078492974889t^{9} + 30468072197731934t^{8} + 2911098632404972t^{7} - 1334901254388946t^{6} - 509752267357253t^{5} - 66312803945016t^{4} - 2748690972216t^{3} - 53468826984t^{2} - 571809780t - 1710720 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ4P4 × ℙ7P7

|λ| = 20.63566 λ ≈ 20.63566; multiplicity 1; Jordan blocks 1 λ ≈ -11.26591 - 5.71244i; multiplicity 1; Jordan blocks 1 λ ≈ -11.26591 + 5.71244i; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 4; Jordan blocks 1, 3 λ ≈ -0.5519202 - 0.09231753i; multiplicity 1; Jordan blocks 1 λ ≈ -0.5519202 + 0.09231753i; multiplicity 1; Jordan blocks 1 4
Sector dimension
9
Spectral radius
$20.6356574$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$20.63566$11
$-11.26591 - 5.71244\mathrm{i}$11
$-11.26591 + 5.71244\mathrm{i}$11
$-1$41, 3
$-0.5519202 - 0.09231753\mathrm{i}$11
$-0.5519202 + 0.09231753\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{4} \left(\lambda^{5} + 3 \lambda^{4} - 303 \lambda^{3} - 3629 \lambda^{2} - 3730 \lambda - 1031\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right)^{3} \left(\lambda^{5} + 3 \lambda^{4} - 303 \lambda^{3} - 3629 \lambda^{2} - 3730 \lambda - 1031\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 16 & 114 & 57 & 156 & 620 & 500 & 2280 \\ 1 & -1 & 4 & 10 & 18 & 2 & 240 & 159 & 160 \\ 1 & 1 & -1 & 38 & 24 & 46 & 111 & 78 & 480 \\ 0 & 1 & 1 & -2 & -3 & 0 & 2 & 6 & 48 \\ 0 & 1 & 0 & 4 & 0 & 7 & 2 & 6 & 96 \\ 0 & 0 & 1 & 1 & 3 & -1 & 44 & 30 & 24 \\ 0 & 0 & 0 & 1 & -\frac{9}{17} & \frac{41}{17} & -\frac{73}{17} & -\frac{15}{17} & \frac{384}{17} \\ 0 & 0 & 0 & 1 & \frac{37}{17} & -\frac{8}{17} & \frac{132}{17} & \frac{39}{17} & -\frac{128}{17} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{15}{8} & \frac{11}{8} & 0\end{pmatrix}\notag\end{equation}$$