Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-66-21-a

·

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

Defining data
Ambient
Fl(1, 4, 7)
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_6ω1ω2ω3ω4ω5ω6
Bundle
3 copies of E^(w6) + E^(2w4) + E^(3w1)
Bundle
3 copies of wedge^2 Q_1^v ot wedge^3 U_2^v ot U_3^v + O(0,2) + O(3,0)
ZL62 label
608.5559C
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4,6) \\ \det\mathcal{E} &\cong \mathcal{O}(3,5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 21
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 66
$\chi(X, \mathrm{T}_X)$ = -49

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 66
$\mathrm{c}_1^2\mathrm{c}_2$ = 108
$\mathrm{c}_2^2$ = 150
$\mathrm{c}_1\mathrm{c}_3$ = -12
$\mathrm{c}_4$ = 84
Hodge diamond
1
00
0 3 0
0 1 1 0
0 7 66 7 0
0 1 1 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 + x2 + 49 0 x0 + 49 x2
$\mathrm{HH}^{ 3 }$ = x1 + 6 0 0 x1 6
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 40 0 0 x1 - x2 + 16 3 21
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 16t + 32t^{2} + 16t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(710)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(707)

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( 21t - 1 \right) \left( 10t + 1 \right) ^{2} \left( 6t + 1 \right) ^{4} \left( 9t^{2} - 42t + 1 \right) \left( 1244203187998608t^{10} + 969914232254952t^{9} + 319210122187710t^{8} + 56084287646799t^{7} + 5676830013363t^{6} + 403899520089t^{5} + 30900236407t^{4} + 1731486533t^{3} + 1448609t^{2} - 1915349t + 2695 \right)D^{6} + \left( 10t + 1 \right) \left( 6t + 1 \right) ^{3} \left( 296294547189988509120t^{15} - 852568760664042013152t^{14} - 857541848050301859000t^{13} - 332041946505878422548t^{12} - 68696429880804161958t^{11} - 8359317912079794015t^{10} - 694587929015349261t^{9} - 56575740319658820t^{8} - 4076027298748182t^{7} - 74217096353292t^{6} + 8295842059496t^{5} + 188672007770t^{4} - 1745787620t^{3} + 102254683t^{2} - 5735267t + 5390 \right)D^{5} + \left( 6t + 1 \right) ^{2} \left( 148147273594994254560000t^{17} - 293073986364842256393600t^{16} - 392686529929472562406560t^{15} - 194647581323773507075248t^{14} - 53591774711687540097192t^{13} - 9257942927561369846004t^{12} - 1098749749767684654402t^{11} - 104235816980389306941t^{10} - 9066153332330152236t^{9} - 622179095652272979t^{8} - 21543763742144448t^{7} + 173173877925194t^{6} + 27253109235072t^{5} - 17607159418t^{4} - 24455417062t^{3} - 99248469t^{2} + 3507228t - 2695 \right)D^{4} + t \left( 6t + 1 \right) \left( 3733311294593855214912000t^{17} - 4907897711326479402432000t^{16} - 8241554354412182140710720t^{15} - 4656979100431200057192000t^{14} - 1457740065524820125378832t^{13} - 289748979998574716444112t^{12} - 39887907965378360101332t^{11} - 4289420398739631308532t^{10} - 411079196697426957591t^{9} - 33571391990109752871t^{8} - 1791289858847622456t^{7} - 33124383454063488t^{6} + 1549453624487022t^{5} + 75671738878366t^{4} + 407527130676t^{3} - 9302489172t^{2} + 129806129t + 22785 \right)D^{3} + 4 t ^{2} \left( 12373260290653920140851200t^{17} - 9499332297890853108264960t^{16} - 22559498419629537977070528t^{15} - 14795873160464545746387552t^{14} - 5298828475323285759638952t^{13} - 1211680373759810830688076t^{12} - 192752955829463584578750t^{11} - 23538570253897018723407t^{10} - 2487050507069838201411t^{9} - 232361990141785212726t^{8} - 16371075175665644448t^{7} - 649240386651672795t^{6} - 2287716035267439t^{5} + 852596650981326t^{4} + 26181998257016t^{3} + 123608678444t^{2} - 553144536t + 18352378 \right)D^{2} + 24 t ^{2} \left( 2239986776756313128947200t^{17} - 1095562341452412673747200t^{16} - 3151763169409421753127552t^{15} - 2063552837837320894044096t^{14} - 721871434031712276513816t^{13} - 159881799232641664767204t^{12} - 24730163894404962047826t^{11} - 3019068011446707256137t^{10} - 330598809752592248541t^{9} - 31595312152241251029t^{8} - 2145227410439358231t^{7} - 74086107137814479t^{6} + 435400318064897t^{5} + 121282533826767t^{4} + 3052468248793t^{3} + 12266104696t^{2} + 50563572t + 909370 \right)D + 48 t ^{2} \left( 457140158521696556928000t^{17} - 137529040186551306816000t^{16} - 519572424995959516346880t^{15} - 342370893293275364607744t^{14} - 117930248384825137456656t^{13} - 25517161607394919467528t^{12} - 3874513057001007906636t^{11} - 477321608250538856766t^{10} - 54208656232939039446t^{9} - 5259153057748201323t^{8} - 343801874556370587t^{7} - 10333547556773770t^{6} + 164187084980060t^{5} + 20398610973459t^{4} + 438348746465t^{3} + 1541438074t^{2} + 15959160t + 70070 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 4, 7)

|λ| = 41.78461 λ ≈ 41.78461; multiplicity 1; Jordan blocks 1 λ ≈ 21; multiplicity 1; Jordan blocks 1 λ ≈ -10; multiplicity 2; Jordan blocks 1, 1 λ ≈ -6; multiplicity 4; Jordan blocks 1, 3 λ ≈ 0.2153903; multiplicity 1; Jordan blocks 1 2 4
Sector dimension
9
Spectral radius
$41.7846097$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$41.78461$11
$21$11
$-10$21, 1
$-6$41, 3
$0.2153903$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 6\right)^{4} \left(\lambda + 10\right)^{2} \left(\lambda - 21\right) \left(\lambda^{2} - 42 \lambda + 9\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 6\right)^{3} \left(\lambda + 10\right) \left(\lambda - 21\right) \left(\lambda^{2} - 42 \lambda + 9\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 132 & 24 & 648 & 1512 & 0 & 864 & 7200 & 15120 \\ 1 & 9 & 2 & 60 & 168 & -4 & 0 & 648 & 864 \\ 1 & 6 & -2 & 132 & 168 & 28 & 648 & 1620 & 6336 \\ 0 & 1 & 1 & 7 & 15 & 2 & 64 & 168 & 648 \\ 0 & 1 & 0 & 2 & 0 & 0 & -4 & 24 & 0 \\ 0 & 0 & 1 & 6 & 6 & -2 & 108 & 144 & 864 \\ 0 & 0 & 0 & 1 & 0 & 1 & 7 & 15 & 108 \\ 0 & 0 & 0 & 1 & 2 & 0 & 2 & 0 & 24 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0\end{pmatrix}\notag\end{equation}$$