Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-352-75-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^(w2) + E^(w3)
Bundle
O(0,1) + Q_1^v ot U_2^v ot U_3^v
ZL62 label
52.21
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,5) \\ \det\mathcal{E} &\cong \mathcal{O}(0,4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 75
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 352
$\chi(X, \mathrm{T}_X)$ = 9

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 352
$\mathrm{c}_1^2\mathrm{c}_2$ = 184
$\mathrm{c}_2^2$ = 96
$\mathrm{c}_1\mathrm{c}_3$ = 60
$\mathrm{c}_4$ = 12
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 4 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 - 9 0 x0 - 9 x1
$\mathrm{HH}^{ 3 }$ = x1 + 55 0 0 x1 - 42 97
$\mathrm{HH}^{ 4 }$ = 75 0 0 0 0 75
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 70t + 210t^{2} + 70t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(82)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(125)

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( 28t^{2} + 4t - 1 \right) \left( 36t^{2} + 4t + 1 \right) \left( 17t^{4} + 21t^{3} - 5t^{2} + 3t + 1 \right) \left( 251843930352t^{9} + 735197705536t^{8} + 556077171420t^{7} + 80569208552t^{6} + 4083789258t^{5} + 1845139314t^{4} + 168300667t^{3} + 17482871t^{2} - 5048t + 4576 \right)D^{6} + \left( 90627549400749312t^{17} + 386423624149339392t^{16} + 558431315789078592t^{15} + 311957830655738432t^{14} + 63030624447945360t^{13} + 22962590773699280t^{12} + 10057438370558444t^{11} + 1750590450673580t^{10} + 217030252171026t^{9} + 57690619761310t^{8} + 12612180333393t^{7} + 1394930603153t^{6} + 82490275375t^{5} + 12583206305t^{4} + 853027055t^{3} + 70303556t^{2} - 42600t + 9152 \right)D^{5} + \left( 755229578339577600t^{17} + 3132797664389379840t^{16} + 4363763482242496640t^{15} + 2356972951939211328t^{14} + 422939093613646112t^{13} + 94692531415970800t^{12} + 33636574775246220t^{11} + 4597805222649428t^{10} + 622385034028250t^{9} + 139157374733368t^{8} + 11764535820177t^{7} + 1141222281824t^{6} + 46152169980t^{5} - 15339053916t^{4} - 741774716t^{3} - 72102383t^{2} - 132696t - 4576 \right)D^{4} + t \left( 3171964229026225920t^{16} + 12869332818813515520t^{15} + 17283118262585152320t^{14} + 8972998144406687424t^{13} + 1543559241800894416t^{12} + 249428248425649104t^{11} + 75386465571654164t^{10} + 8949674764402956t^{9} + 1760943296767918t^{8} + 390290503562146t^{7} + 51054643261595t^{6} + 5368701900159t^{5} + 211666410649t^{4} + 3809435389t^{3} + 120393609t^{2} + 2343386t + 936 \right)D^{3} + 2 t ^{2} \left( 3504265243495640064t^{15} + 13973767288104056832t^{14} + 18156007314659159008t^{13} + 9026695439711311168t^{12} + 1541006463690563496t^{11} + 217580305585598336t^{10} + 62659695352706184t^{9} + 7254867358401252t^{8} + 1627264853654674t^{7} + 263790081417104t^{6} + 30488905855161t^{5} + 3221457358729t^{4} + 96499845170t^{3} + 1985637861t^{2} + 79392694t + 936144 \right)D^{2} + 4 t ^{2} \left( 1903178537415735552t^{15} + 7492176782043432192t^{14} + 9472282726998988512t^{13} + 4516575920021979328t^{12} + 774393560627867432t^{11} + 107860745269668992t^{10} + 31586243121756556t^{9} + 3710357703226992t^{8} + 802711871584440t^{7} + 96222777237262t^{6} + 9895165769556t^{5} + 1062791852330t^{4} + 24337822705t^{3} + 688685084t^{2} + 25101408t + 201344 \right)D + 32 t ^{2} \left( 97100945786517120t^{15} + 378909560173576320t^{14} + 469577885019359760t^{13} + 216390486297008688t^{12} + 37343277987387316t^{11} + 5356705013429340t^{10} + 1610325136723844t^{9} + 191061367318260t^{8} + 38496459602838t^{7} + 3638877050270t^{6} + 335966102520t^{5} + 36900682260t^{4} + 674280875t^{3} + 27446110t^{2} + 808132t + 4576 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 3, 4)

|λ| = 7.656854 λ ≈ 7.656854; multiplicity 1; Jordan blocks 1 λ ≈ -2 - 5.656854i; multiplicity 1; Jordan blocks 1 λ ≈ -2 + 5.656854i; multiplicity 1; Jordan blocks 1 λ ≈ -4.799605; multiplicity 2; Jordan blocks 1, 1 λ ≈ -3.656854; multiplicity 1; Jordan blocks 1 λ ≈ 1.234552 - 1.940705i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 1.234552 + 1.940705i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -0.6694996; multiplicity 2; Jordan blocks 1, 1 2 2 2 2
Sector dimension
12
Spectral radius
$7.65685425$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$7.656854$11
$-2 - 5.656854\mathrm{i}$11
$-2 + 5.656854\mathrm{i}$11
$-4.799605$21, 1
$-3.656854$11
$1.234552 - 1.940705\mathrm{i}$21, 1
$1.234552 + 1.940705\mathrm{i}$21, 1
$-0.6694996$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda^{2} + 4 \lambda + 36\right) \left(\lambda^{2} - 4 \lambda - 28\right) \left(\lambda^{4} + 3 \lambda^{3} - 5 \lambda^{2} + 21 \lambda + 17\right)^{2}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda^{2} + 4 \lambda + 36\right) \left(\lambda^{2} - 4 \lambda - 28\right) \left(\lambda^{4} + 3 \lambda^{3} - 5 \lambda^{2} + 21 \lambda + 17\right)\notag\end{equation}$$

Quantum multiplication matrix

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