Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-148-36-a

·

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

Defining data
Ambient
ℙ2P2 × Fl(1, 3, 4)
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_5 (marked)ω1ω2ω3ω4ω5
Bundle
E^(w3 + w5) + E^(w1 + w4)
Bundle
O_2(1,1) + O_1(1) box U_2,2^v ot U_2,3^v
ZL62 label
2034.hb
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3,3) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2,2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 36
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 148
$\chi(X, \mathrm{T}_X)$ = -8

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 148
$\mathrm{c}_1^2\mathrm{c}_2$ = 124
$\mathrm{c}_2^2$ = 116
$\mathrm{c}_1\mathrm{c}_3$ = 44
$\mathrm{c}_4$ = 20
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 12 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 + 8 0 x0 + 8 x1
$\mathrm{HH}^{ 3 }$ = x1 + 25 0 0 x1 + 7 18
$\mathrm{HH}^{ 4 }$ = 36 0 0 0 0 36
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 31t + 84t^{2} + 31t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(431)
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( 5t + 1 \right) \left( t + 1 \right) ^{2} \left( 32t^{2} + 8t + 1 \right) \left( 256t^{3} + 32t^{2} + 11t - 1 \right) \left( 30081024000t^{14} + 150405120000t^{13} + 361607528448t^{12} + 605017724928t^{11} + 711323608576t^{10} + 477821488576t^{9} + 111591347392t^{8} - 61737329568t^{7} - 45719384480t^{6} - 10003600770t^{5} - 837318184t^{4} - 17985686t^{3} + 1894642t^{2} + 13208t - 851 \right)D^{6} + \left( t + 1 \right) \left( 17249662402560000t^{21} + 108118419701760000t^{20} + 328693173222113280t^{19} + 678595928084447232t^{18} + 1021854822423330816t^{17} + 1056375573855404032t^{16} + 676285263992504320t^{15} + 196820991784050688t^{14} - 50415820508960256t^{13} - 72274904095759680t^{12} - 31873185790907776t^{11} - 8102216567702048t^{10} - 1297125903210192t^{9} - 128725164795746t^{8} - 7951212975108t^{7} - 544179906228t^{6} - 68368723996t^{5} - 5299258112t^{4} - 113696504t^{3} + 9008937t^{2} + 57938t - 2553 \right)D^{5} + \left( 97029351014400000t^{22} + 683402361569280000t^{21} + 2325742145136230400t^{20} + 5330801213728358400t^{19} + 8984687937835761664t^{18} + 10736221430112649216t^{17} + 8379382823945256960t^{16} + 3527298297431056384t^{15} - 54427777311763968t^{14} - 982155330269844160t^{13} - 588718368803160320t^{12} - 184554716543065728t^{11} - 35211162674802432t^{10} - 4148135707220646t^{9} - 267395997219636t^{8} - 1340870650602t^{7} + 1734875770046t^{6} + 203883841446t^{5} + 10567977947t^{4} + 226752165t^{3} - 8759917t^{2} - 21529t + 1702 \right)D^{4} + t \left( 280307014041600000t^{21} + 1928419847700480000t^{20} + 6450064287753830400t^{19} + 14791027964636037120t^{18} + 25030877015489642496t^{17} + 29386142078846042112t^{16} + 21582681760244645888t^{15} + 7437279233950449664t^{14} - 1801753719886253568t^{13} - 3255342501279434432t^{12} - 1614779628118731968t^{11} - 435239479334763168t^{10} - 69693350129361040t^{9} - 6485141005827678t^{8} - 269445470425730t^{7} + 6812685620672t^{6} + 1135870818664t^{5} + 1523226420t^{4} - 3590487920t^{3} - 105106453t^{2} - 1335294t + 2921 \right)D^{3} + t \left( 437710183464960000t^{21} + 2959264293322752000t^{20} + 9774830884245995520t^{19} + 22530868977843830784t^{18} + 38434978504292958208t^{17} + 44523755774262509568t^{16} + 30965389221798363136t^{15} + 8520054892852871168t^{14} - 4607776347127490048t^{13} - 5432494795927269760t^{12} - 2361062721811258944t^{11} - 564550903199326944t^{10} - 78572795289460272t^{9} - 6078393041947884t^{8} - 176823793034948t^{7} + 6920134368936t^{6} + 272607187972t^{5} - 73512258936t^{4} - 5200800802t^{3} - 101074669t^{2} - 673178t + 851 \right)D^{2} + 2 t ^{2} \left( 174652831825920000t^{20} + 1165738345758720000t^{19} + 3816218920809922560t^{18} + 8863834869180923904t^{17} + 15271234283021795328t^{16} + 17525016361352298496t^{15} + 11636477823908839424t^{14} + 2495706422599491584t^{13} - 2363363542514575872t^{12} - 2251128947126873088t^{11} - 884606981199900992t^{10} - 191707346151421408t^{9} - 23841899353727072t^{8} - 1613101960939176t^{7} - 42236922328240t^{6} - 314378874972t^{5} - 278126557376t^{4} - 39755867818t^{3} - 1653599118t^{2} - 27944961t - 68376 \right)D + 192 t ^{3} \left( 577555660800000t^{19} + 3819086807040000t^{18} + 12422887007846400t^{17} + 29092948617461760t^{16} + 50627945629089792t^{15} + 57743634769035264t^{14} + 36945038009373696t^{13} + 6066451315470336t^{12} - 9113307868786112t^{11} - 7632369790615456t^{10} - 2774289715794904t^{9} - 554483360687668t^{8} - 63095411561120t^{7} - 3943548727462t^{6} - 127962800334t^{5} - 10407641961t^{4} - 1753683920t^{3} - 143853982t^{2} - 4369458t - 68931 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Fl(1, 3, 4)

|λ| = 14.44287 λ ≈ 14.44287; multiplicity 1; Jordan blocks 1 λ ≈ -4 - 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -4 + 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -5; multiplicity 1; Jordan blocks 1 λ ≈ -1.721437 - 3.84209i; multiplicity 1; Jordan blocks 1 λ ≈ -1.721437 + 3.84209i; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ 0; multiplicity 3; Jordan blocks 1, 1, 1 2 2 3 3
Sector dimension
14
Spectral radius
$14.4428745$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$14.44287$11
$-4 - 4\mathrm{i}$21, 1
$-4 + 4\mathrm{i}$21, 1
$-5$11
$-1.721437 - 3.84209\mathrm{i}$11
$-1.721437 + 3.84209\mathrm{i}$11
$-1$31, 1, 1
$0$31, 1, 1

Characteristic polynomial

$$\begin{equation}\lambda^{3} \left(\lambda + 1\right)^{3} \left(\lambda + 5\right) \left(\lambda^{2} + 8 \lambda + 32\right)^{2} \left(\lambda^{3} - 11 \lambda^{2} - 32 \lambda - 256\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right) \left(\lambda + 5\right) \left(\lambda^{2} + 8 \lambda + 32\right) \left(\lambda^{3} - 11 \lambda^{2} - 32 \lambda - 256\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 6 & 6 & 24 & 24 & 12 & 24 & 18 & 18 & 88 & 64 & 64 & 40 \\ 1 & -1 & 1 & 1 & 4 & 4 & 0 & 8 & 6 & 6 & 18 & 12 & 12 & 8 \\ 1 & 1 & -1 & 1 & 2 & 6 & 4 & 4 & 4 & 4 & 24 & 24 & 12 & 12 \\ 1 & 1 & 1 & -1 & 6 & 2 & 4 & 4 & 4 & 4 & 24 & 12 & 24 & 12 \\ 0 & 1 & 1 & 0 & -2 & 1 & 0 & 1 & 0 & 2 & 2 & 4 & 0 & 3 \\ 0 & 1 & 0 & 1 & 1 & -2 & 0 & 1 & 2 & 0 & 2 & 0 & 4 & 3 \\ 0 & 1 & 0 & 0 & 1 & 1 & -1 & 0 & 0 & 0 & 8 & 6 & 6 & 0 \\ 0 & 0 & 1 & 1 & 1 & 1 & 0 & -2 & -2 & -2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 2 & 6 & 2 & 6 & 3 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 2 & 0 & 6 & 6 & 2 & 3 \\ 0 & 0 & 0 & 0 & 3 & 3 & 4 & -2 & -2 & -2 & -2 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & -1 & -2 & 3 & \frac{7}{2} & \frac{3}{2} & 2 & -\frac{1}{2} & \frac{3}{2} & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & -2 & 3 & \frac{3}{2} & \frac{7}{2} & 2 & \frac{3}{2} & -\frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 6 & 6 & 0\end{pmatrix}\notag\end{equation}$$