Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-288-63-f

·

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

Defining data
Ambient
ℙ2P2 × ℙ2P2 × Gr(2, 5)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6 (marked)node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w6) + E^(w3 + w6) + E^(w2 + w5)
Bundle
O_3(1) + O_2(1) box O_3(1) + Q_1 box U_3^v
ZL62 label
202041.5wIw4
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 63
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 288
$\chi(X, \mathrm{T}_X)$ = 2

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 288
$\mathrm{c}_1^2\mathrm{c}_2$ = 168
$\mathrm{c}_2^2$ = 99
$\mathrm{c}_1\mathrm{c}_3$ = 54
$\mathrm{c}_4$ = 15
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 7 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 + x1 - 2 0 x0 - 2 x1
$\mathrm{HH}^{ 3 }$ = x1 + 47 0 0 x1 - 25 72
$\mathrm{HH}^{ 4 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 58t + 170t^{2} + 58t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(175)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(247)

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) \left( 404t^{3} + 58t^{2} - 3t - 1 \right) \left( 43t^{4} + 36t^{3} + 8t^{2} + t + 1 \right) \left( 1255327334400t^{13} + 6469100156160t^{12} + 6175920393984t^{11} + 3261478460416t^{10} + 1579554084792t^{9} + 847903768096t^{8} + 177640276063t^{7} + 13824608906t^{6} - 3503643234t^{5} + 183652152t^{4} + 26579084t^{3} + 679256t^{2} + 18924t + 432 \right)D^{6} + \left( 915916951034265600t^{22} + 6042702520067297280t^{21} + 11090028742620162048t^{20} + 9549015424622106624t^{19} + 4962103620614066240t^{18} + 2146717557785372736t^{17} + 828467300958232216t^{16} + 68494731701810048t^{15} - 126953881305118720t^{14} - 60723050693522085t^{13} - 11203218176936566t^{12} - 708144827055907t^{11} - 60443857695643t^{10} - 50829104748703t^{9} - 6789620561995t^{8} - 2087484061333t^{7} - 169149683122t^{6} + 17142049374t^{5} - 1388813752t^{4} - 139478440t^{3} - 2797628t^{2} - 54180t - 864 \right)D^{5} + \left( 7632641258618880000t^{22} + 50132028287392665600t^{21} + 87344997814080061440t^{20} + 74419248868563724288t^{19} + 42049121122810403008t^{18} + 21596315482989646016t^{17} + 9450857935741489960t^{16} + 1773114250966760240t^{15} - 436844354875155440t^{14} - 281566238625878385t^{13} - 42970767218411412t^{12} + 4094442298041199t^{11} + 1288445604192120t^{10} + 55242089801438t^{9} + 25338976253973t^{8} + 2206231640233t^{7} + 125158028900t^{6} - 58505774134t^{5} - 372840836t^{4} + 110130788t^{3} + 2557972t^{2} + 60564t + 432 \right)D^{4} + t \left( 32057093286199296000t^{21} + 209815892853594316800t^{20} + 348602300091052922880t^{19} + 292787574941058263040t^{18} + 173316394182530730176t^{17} + 97010988607541282496t^{16} + 43960929678055179432t^{15} + 9586960042809928864t^{14} - 517258134520836608t^{13} - 726672032557821443t^{12} - 104023448838007118t^{11} + 14253291065711155t^{10} + 2565628281394080t^{9} - 154210496774217t^{8} - 29842887006277t^{7} - 795862379327t^{6} + 58673472870t^{5} - 3281188214t^{4} - 625919836t^{3} - 27258352t^{2} - 460660t - 108 \right)D^{3} + t ^{2} \left( 70830910879983206400t^{20} + 462344581288529909760t^{19} + 737527838440446541824t^{18} + 610277024325040451584t^{17} + 371552224509874114048t^{16} + 218989962842870321408t^{15} + 99649384781466589200t^{14} + 22708084782512565808t^{13} + 251370953336441120t^{12} - 1089897918722382016t^{11} - 137662545579651668t^{10} + 24566458406351406t^{9} + 2669497933969080t^{8} - 386729436946851t^{7} - 62777709453514t^{6} + 2495245875404t^{5} + 268282016456t^{4} + 140397080t^{3} - 658815760t^{2} - 27711108t - 368736 \right)D^{2} + 4 t ^{2} \left( 19234255971719577600t^{20} + 125301992340090209280t^{19} + 193508061278411593728t^{18} + 158003845152807358464t^{17} + 97939387952062055744t^{16} + 59681992759181363136t^{15} + 26942922161143346768t^{14} + 6188705028837549864t^{13} + 226554681559659952t^{12} - 227100656965570021t^{11} - 24016781509536326t^{10} + 5293505339267224t^{9} + 325338259029768t^{8} - 92050079932572t^{7} - 12676019839072t^{6} + 909309543244t^{5} + 59406053952t^{4} + 18368864t^{3} - 98582048t^{2} - 3905040t - 39168 \right)D + 48 t ^{2} \left( 654226393595904000t^{20} + 4256262680449459200t^{19} + 6422744285848627200t^{18} + 5191098175591901184t^{17} + 3254623782309991616t^{16} + 2025921499115111232t^{15} + 905278631863913504t^{14} + 207147317015192616t^{13} + 9662926821879712t^{12} - 6585435163306459t^{11} - 571797133551566t^{10} + 151587282229882t^{9} + 5020197840024t^{8} - 2584025881674t^{7} - 309460420204t^{6} + 29650834300t^{5} + 1495786128t^{4} - 6303112t^{3} - 2158736t^{2} - 79032t - 576 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2 × Gr(2, 5)

|λ| = 8.802676 λ ≈ 8.802676; multiplicity 1; Jordan blocks 1 λ ≈ -5.901338 - 3.327062i; multiplicity 1; Jordan blocks 1 λ ≈ -5.901338 + 3.327062i; multiplicity 1; Jordan blocks 1 λ ≈ 1.206542 - 3.383479i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 1.206542 + 3.383479i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 2; multiplicity 1; Jordan blocks 1 λ ≈ -1.706542 - 0.6481502i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1.706542 + 0.6481502i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1; multiplicity 1; Jordan blocks 1 2 2 2 2
Sector dimension
13
Spectral radius
$8.80267607$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$8.802676$11
$-5.901338 - 3.327062\mathrm{i}$11
$-5.901338 + 3.327062\mathrm{i}$11
$1.206542 - 3.383479\mathrm{i}$21, 1
$1.206542 + 3.383479\mathrm{i}$21, 1
$2$11
$-1.706542 - 0.6481502\mathrm{i}$21, 1
$-1.706542 + 0.6481502\mathrm{i}$21, 1
$-1$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda - 2\right) \left(\lambda^{3} + 3 \lambda^{2} - 58 \lambda - 404\right) \left(\lambda^{4} + \lambda^{3} + 8 \lambda^{2} + 36 \lambda + 43\right)^{2}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda - 2\right) \left(\lambda^{3} + 3 \lambda^{2} - 58 \lambda - 404\right) \left(\lambda^{4} + \lambda^{3} + 8 \lambda^{2} + 36 \lambda + 43\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 2 & 2 & 0 & 6 & 3 & 3 & 3 & 4 & 4 & 4 & 10 \\ 1 & 0 & 0 & 2 & 2 & -4 & -2 & 0 & 0 & 3 & 3 & 0 & -4 \\ 2 & 0 & 0 & 1 & 2 & 2 & 2 & 2 & 0 & 6 & 0 & 3 & 0 \\ 1 & 0 & 0 & -1 & 0 & 4 & 2 & 2 & 2 & 0 & 0 & 0 & 4 \\ 0 & 2 & 1 & 0 & 0 & 1 & 0 & 2 & 0 & -4 & 0 & -2 & 0 \\ 0 & 1 & 0 & 3 & 0 & -1 & 0 & 0 & 0 & 2 & 2 & 0 & 0 \\ 0 & 1 & 0 & -1 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 3 \\ 0 & 0 & 1 & 2 & 0 & 0 & 0 & -1 & 0 & 4 & 0 & 2 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 2 & 2 & 3 \\ 0 & 0 & 0 & 0 & 1 & \frac{7}{2} & \frac{1}{2} & 4 & \frac{1}{2} & -1 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 2 & -\frac{3}{2} & -\frac{1}{2} & 3 & \frac{5}{2} & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 2 & -3 & -1 & 2 & 0 & 0 & -2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 3 & 3 & 0\end{pmatrix}\notag\end{equation}$$