Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-98-27-a

·

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

Defining data
Ambient
ℙ3P3 × ℙ9P9
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 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_11node 12: fundamental weight ω_12ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10ω11ω12
Bundle
E^(2w4) + E^(w1 + w4) + 2 copies of E^(w3 + w4)
Bundle
O_2(2) + O_1(1) box O_2(1) + 2 copies of Q_1 box O_2(1)
ZL62 label
30900.Fw0Lw0Lw02
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;10) \\ \det\mathcal{E} &\cong \mathcal{O}(3;9) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 27
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 98
$\chi(X, \mathrm{T}_X)$ = -35

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 98
$\mathrm{c}_1^2\mathrm{c}_2$ = 116
$\mathrm{c}_2^2$ = 134
$\mathrm{c}_1\mathrm{c}_3$ = 4
$\mathrm{c}_4$ = 52
Hodge diamond
1
00
0 2 0
0 0 0 0
0 3 40 3 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 + x2 + 35 0 x0 + 35 x2
$\mathrm{HH}^{ 3 }$ = x1 + 13 0 0 x1 13
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 29 0 0 x1 - x2 + 2 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 22t + 52t^{2} + 22t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(495)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(545)

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) \left( 2t + 1 \right) ^{4} \left( 28t^{2} + 20t - 1 \right) \left( 164t^{2} + 12t + 1 \right) \left( 46808238080t^{10} + 256147299840t^{9} + 467552276224t^{8} + 428083517568t^{7} + 223034897280t^{6} + 68045034144t^{5} + 12112286752t^{4} + 1390238136t^{3} + 128013872t^{2} + 788904t + 4851 \right)D^{6} + 2 \left( 2t + 1 \right) ^{3} \left( 4513812014530560t^{16} + 33268232844820480t^{15} + 95550306106998784t^{14} + 149472417820291072t^{13} + 145047170230969344t^{12} + 92772370512256000t^{11} + 40175534105082624t^{10} + 11898607415267840t^{9} + 2438676214889728t^{8} + 359058620259776t^{7} + 39155538316512t^{6} + 2675439493808t^{5} + 76163945412t^{4} + 2722227732t^{3} + 253039159t^{2} + 1163952t + 4851 \right)D^{5} + \left( 2t + 1 \right) ^{2} \left( 150460400484352000t^{17} + 1137298976083148800t^{16} + 3362857778429460480t^{15} + 5480583369087795200t^{14} + 5628179086648418304t^{13} + 3883224683304620032t^{12} + 1857402117344152576t^{11} + 625173025719461888t^{10} + 149927947293678336t^{9} + 26265242343495936t^{8} + 3456368524207104t^{7} + 320405263329344t^{6} + 15304419313792t^{5} + 132481964784t^{4} - 3539361972t^{3} - 462669392t^{2} - 2035815t - 4851 \right)D^{4} + 2 t \left( 2t + 1 \right) \left( 631933682034278400t^{17} + 4937943887911321600t^{16} + 15224466466204712960t^{15} + 26181719099568373760t^{14} + 28765559641138974720t^{13} + 21577456338827931648t^{12} + 11435923884244391936t^{11} + 4359359092659332096t^{10} + 1209807485916241152t^{9} + 248590183836360448t^{8} + 38704968742295424t^{7} + 4525734092569536t^{6} + 350638174271728t^{5} + 13147283850576t^{4} + 150218093320t^{3} + 1352179020t^{2} + 8755365t + 5049 \right)D^{3} + 4 t ^{2} \left( 1396272516494786560t^{17} + 11347160269191741440t^{16} + 36826320174363066368t^{15} + 67421930868876058624t^{14} + 79807347732787740672t^{13} + 65368600126607480832t^{12} + 38424152375034698752t^{11} + 16534422335890830848t^{10} + 5272845656542641664t^{9} + 1261477042807516800t^{8} + 230142347562253376t^{7} + 32246243927770272t^{6} + 3305151468076192t^{5} + 210810777580856t^{4} + 6091360398684t^{3} + 69420279902t^{2} + 862127572t + 3655155 \right)D^{2} + 16 t ^{2} \left( 379160209220567040t^{17} + 3029306600017838080t^{16} + 9554501084107104256t^{15} + 16926021946059886592t^{14} + 19337199723094917120t^{13} + 15253477700769682432t^{12} + 8615798560264318464t^{11} + 3555670837442970624t^{10} + 1086722885324307584t^{9} + 249689168679473728t^{8} + 43934357237200128t^{7} + 5923251329179280t^{6} + 569814133853592t^{5} + 32188817275220t^{4} + 753437784132t^{3} + 9141886350t^{2} + 110114328t + 354123 \right)D + 96 t ^{2} \left( 25793211511603200t^{17} + 203683834066534400t^{16} + 629634805677670400t^{15} + 1089557048892129280t^{14} + 1213319519813709824t^{13} + 931089466209172480t^{12} + 510603469332568576t^{11} + 204215032167982848t^{10} + 60444554480115456t^{9} + 13471596611425664t^{8} + 2306046836541696t^{7} + 301233852079040t^{6} + 27413524618544t^{5} + 1389890103220t^{4} + 26995745818t^{3} + 354019389t^{2} + 3505447t + 8085 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × ℙ9P9

|λ| = 21.31371 λ ≈ 21.31371; multiplicity 1; Jordan blocks 1 λ ≈ -6 - 11.31371i; multiplicity 1; Jordan blocks 1 λ ≈ -6 + 11.31371i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 4; Jordan blocks 1, 3 λ ≈ -1.313708; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 1; Jordan blocks 1 4
Sector dimension
9
Spectral radius
$21.3137085$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$21.31371$11
$-6 - 11.31371\mathrm{i}$11
$-6 + 11.31371\mathrm{i}$11
$-2$41, 3
$-1.313708$11
$-1$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda + 2\right)^{4} \left(\lambda^{2} + 12 \lambda + 164\right) \left(\lambda^{2} - 20 \lambda - 28\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda + 2\right)^{3} \left(\lambda^{2} + 12 \lambda + 164\right) \left(\lambda^{2} - 20 \lambda - 28\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 12 & 168 & 84 & 264 & 1152 & 576 & 3200 \\ 1 & 0 & 6 & -8 & -4 & -8 & 144 & 96 & 720 \\ 1 & 0 & 0 & 28 & 20 & 36 & 348 & 180 & 720 \\ 0 & 1 & 1 & 0 & -2 & 6 & -32 & -16 & 60 \\ 0 & 1 & 0 & 6 & 1 & 10 & 12 & 4 & 60 \\ 0 & 0 & 1 & 0 & 1 & -2 & 40 & 24 & 120 \\ 0 & 0 & 0 & 1 & -\frac{1}{4} & 3 & -\frac{9}{2} & -\frac{3}{2} & 15 \\ 0 & 0 & 0 & 1 & \frac{7}{4} & -1 & \frac{27}{2} & \frac{9}{2} & -5 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{8}{5} & \frac{4}{5} & 0\end{pmatrix}\notag\end{equation}$$