Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-192-45-b

·

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

Defining data
Ambient
Fl(1, 2, 3) × ℙ2P2
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
E^(w1 + w2 + w3)
Bundle
O_1(1,1) box O_2(1)
ZL62 label
2022.x4200
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,2;3) \\ \det\mathcal{E} &\cong \mathcal{O}(1,1;1) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 45
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 192
$\chi(X, \mathrm{T}_X)$ = -7

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 192
$\mathrm{c}_1^2\mathrm{c}_2$ = 144
$\mathrm{c}_2^2$ = 102
$\mathrm{c}_1\mathrm{c}_3$ = 48
$\mathrm{c}_4$ = 18
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 10 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 + 7 0 x0 + 7 x1
$\mathrm{HH}^{ 3 }$ = x1 + 35 0 0 x1 + 2 33
$\mathrm{HH}^{ 4 }$ = 45 0 0 0 0 45
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 40t + 110t^{2} + 40t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(397)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(470)

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( 2t + 1 \right) \left( 79t^{4} + 69t^{3} + 37t^{2} + 11t + 1 \right) \left( 704t^{4} + 192t^{3} - 64t^{2} + 16t - 1 \right) \left( 442394749056t^{12} + 2196710938080t^{11} + 1367579189880t^{10} + 300152261816t^{9} + 363131503254t^{8} + 376953209662t^{7} + 131319709863t^{6} + 18654436700t^{5} + 1067510103t^{4} + 120535461t^{3} + 7498534t^{2} + 140703t - 4928 \right)D^{6} + \left( 1033377507266936832t^{21} + 6752652341648867328t^{20} + 11499242790121792512t^{19} + 9854927599635595776t^{18} + 6119775708652912640t^{17} + 4182091905780031104t^{16} + 3083217398595661136t^{15} + 1820684240092921240t^{14} + 767765568927634452t^{13} + 225410814189141730t^{12} + 45247456662261444t^{11} + 6125362914881329t^{10} + 570172839760257t^{9} + 45763532656408t^{8} + 6253880155080t^{7} + 1315511706878t^{6} + 169867342489t^{5} + 8746933909t^{4} + 741415512t^{3} + 35398270t^{2} + 548028t - 14784 \right)D^{5} + \left( 8611479227224473600t^{21} + 55740953237339996160t^{20} + 86451295711433051136t^{19} + 65763190959426771200t^{18} + 38992371505490305216t^{17} + 30173537408809686176t^{16} + 23518301026620168448t^{15} + 13091803065339891120t^{14} + 4936331124243601932t^{13} + 1272244104547409818t^{12} + 224808830533188566t^{11} + 27265432314540297t^{10} + 2370954196200223t^{9} + 155119656147887t^{8} - 670765878748t^{7} - 2226565434191t^{6} - 298661910305t^{5} - 13100578282t^{4} - 1021286259t^{3} - 35083883t^{2} - 483579t + 9856 \right)D^{4} + t \left( 36168212754342789120t^{20} + 232359209884706119680t^{19} + 331556291730353556480t^{18} + 227413930398883144704t^{17} + 134348206214962350464t^{16} + 118063005629285729088t^{15} + 92962516509334646832t^{14} + 48121880754752151416t^{13} + 16315586496831221748t^{12} + 3736419426181428438t^{11} + 594446702890783204t^{10} + 69215482750294627t^{9} + 6776905291602215t^{8} + 608704122621912t^{7} + 39385228911013t^{6} + 1411372990724t^{5} - 47098230623t^{4} - 6293761343t^{3} - 121399441t^{2} - 5676618t + 16478 \right)D^{3} + t \left( 79914527228643115008t^{20} + 510440150565239894016t^{19} + 678522268969668366336t^{18} + 426822615329891774720t^{17} + 258584351595721337920t^{16} + 252270306447381577376t^{15} + 195783813166769608976t^{14} + 94508193467033677336t^{13} + 29216725171067338496t^{12} + 6056664800968901104t^{11} + 890070833108764900t^{10} + 103424567764239870t^{9} + 11256551889078900t^{8} + 1110927907812925t^{7} + 69615409465604t^{6} + 1570460037048t^{5} - 91700587238t^{4} - 10726837475t^{3} - 118106450t^{2} - 2776352t + 4928 \right)D^{2} + 2 t ^{2} \left( 43401855305211346944t^{19} + 276042516410544377856t^{18} + 346787307424779761664t^{17} + 203649838365263994624t^{16} + 128135635680778981312t^{15} + 134865849664850335008t^{14} + 102340893307525659920t^{13} + 46565856120704434536t^{12} + 13354167034308905720t^{11} + 2558388252884505484t^{10} + 356654141678742068t^{9} + 42356491566667908t^{8} + 4933007326190754t^{7} + 492226683243005t^{6} + 29142760704766t^{5} + 449958427290t^{4} - 36747330512t^{3} - 3962043855t^{2} - 36458992t - 280576 \right)D + 96 t ^{3} \left( 369063395452477440t^{18} + 2340525442142376960t^{17} + 2823372337058118144t^{16} + 1576906027938581184t^{15} + 1028679938347315408t^{14} + 1138262537918161392t^{13} + 846390921936981476t^{12} + 368444569624262228t^{11} + 99892318984271798t^{10} + 18051172122089177t^{9} + 2436648418701366t^{8} + 297419819753659t^{7} + 35912080596988t^{6} + 3510764244011t^{5} + 196439839278t^{4} + 2367685316t^{3} - 233948476t^{2} - 22724139t - 199584 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 3) × ℙ2P2

|λ| = 12.4648 λ ≈ 12.4648; multiplicity 1; Jordan blocks 1 λ ≈ -6.799605; multiplicity 2; Jordan blocks 1, 1 λ ≈ 2.706033 - 4.771747i; multiplicity 1; Jordan blocks 1 λ ≈ 2.706033 + 4.771747i; multiplicity 1; Jordan blocks 1 λ ≈ -2.6695; multiplicity 2; Jordan blocks 1, 1 λ ≈ -0.7654478 - 1.940705i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -0.7654478 + 1.940705i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ -1.876867; multiplicity 1; Jordan blocks 1 2 2 2 2
Sector dimension
13
Spectral radius
$12.4648018$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$12.4648$11
$-6.799605$21, 1
$2.706033 - 4.771747\mathrm{i}$11
$2.706033 + 4.771747\mathrm{i}$11
$-2.6695$21, 1
$-0.7654478 - 1.940705\mathrm{i}$21, 1
$-0.7654478 + 1.940705\mathrm{i}$21, 1
$-2$11
$-1.876867$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda^{4} + 11 \lambda^{3} + 37 \lambda^{2} + 69 \lambda + 79\right)^{2} \left(\lambda^{4} - 16 \lambda^{3} + 64 \lambda^{2} - 192 \lambda - 704\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda^{4} + 11 \lambda^{3} + 37 \lambda^{2} + 69 \lambda + 79\right) \left(\lambda^{4} - 16 \lambda^{3} + 64 \lambda^{2} - 192 \lambda - 704\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 6 & 6 & 2 & 6 & 3 & 6 & 3 & 6 & 4 & 12 & 24 & 30 \\ 1 & -1 & 2 & 0 & 2 & 2 & 2 & 0 & 2 & 0 & 3 & 3 & 8 \\ 1 & 2 & -1 & 0 & 2 & 0 & 2 & 2 & 2 & 0 & 0 & 6 & 8 \\ 2 & 1 & 1 & 0 & 4 & 6 & 4 & 6 & 0 & 6 & 6 & 0 & 4 \\ 0 & 1 & 2 & 0 & -1 & 0 & 1 & 0 & 0 & 0 & 2 & 2 & 3 \\ 0 & 2 & 0 & 1 & 1 & -1 & 0 & 2 & 0 & 2 & 2 & 0 & 0 \\ 0 & 2 & 1 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 4 & 3 \\ 0 & 0 & 2 & 1 & 0 & 2 & 1 & -1 & 0 & 2 & 2 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 1 & 0 & 1 & 0 & 2 & 4 & 6 & 6 \\ 0 & 0 & 0 & 0 & 1 & 3 & 4 & -\frac{3}{2} & -\frac{3}{2} & -\frac{1}{2} & \frac{3}{2} & -3 & -1 \\ 0 & 0 & 0 & 0 & 2 & 0 & -1 & \frac{9}{2} & \frac{3}{2} & \frac{1}{2} & -\frac{3}{2} & 3 & 3 \\ 0 & 0 & 0 & 0 & 0 & 3 & 1 & \frac{3}{2} & \frac{3}{2} & \frac{1}{2} & \frac{3}{2} & 0 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 3 & 0\end{pmatrix}\notag\end{equation}$$