Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-38-15-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ5P5
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_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(w1 + 3w3) + E^(w2 + w3)
Bundle
O_1(1) box O_2(3) + Q_1 box O_2(1)
ZL62 label
2050.w6x312
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;6) \\ \det\mathcal{E} &\cong \mathcal{O}(2;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 15
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 38
$\chi(X, \mathrm{T}_X)$ = -84

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 38
$\mathrm{c}_1^2\mathrm{c}_2$ = 92
$\mathrm{c}_2^2$ = 213
$\mathrm{c}_1\mathrm{c}_3$ = -58
$\mathrm{c}_4$ = 191
Hodge diamond
1
00
0 2 0
0 0 0 0
0 21 143 21 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x2 + 84 0 x0 + 84 x2
$\mathrm{HH}^{ 3 }$ = x1 + 3 0 0 x1 3
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 108 0 0 x1 - x2 + 93 0 15
$\mathrm{HH}^{ 5 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 6 }$ = 1 0 1 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 10t + 16t^{2} + 10t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(725)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(718)

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( 6t + 1 \right) ^{4} \left( 1658312t^{5} + 545972t^{4} + 47238t^{3} + 699t^{2} + 48t - 1 \right) \left( 38138001529244544t^{10} + 33110028828764736t^{9} + 12790742054867232t^{8} + 2939786183480720t^{7} + 447128443722968t^{6} + 46406982141532t^{5} + 3274313544854t^{4} + 155596474483t^{3} + 4379249129t^{2} + 13196717t + 9405 \right)D^{6} + \left( 6t + 1 \right) ^{3} \left( 7968832904587536859465728t^{16} + 9877522376545455156931584t^{15} + 5597427573053254046651904t^{14} + 1933434070207989437060352t^{13} + 456632702120453140706048t^{12} + 78000329256856271079808t^{11} + 9893297431836172896448t^{10} + 940886266251226557344t^{9} + 67072922502266573032t^{8} + 3522553670426755868t^{7} + 128321422715145438t^{6} + 2881253393877713t^{5} + 40088832102054t^{4} + 646026789038t^{3} + 17438710082t^{2} + 39872301t + 18810 \right)D^{5} + \left( 6t + 1 \right) ^{2} \left( 398441645229376842973286400t^{17} + 520584045009427031881789440t^{16} + 313339711707052223543599104t^{15} + 115879787461987053872941056t^{14} + 29549845361287177614667008t^{13} + 5501999438017578848822016t^{12} + 769437799671522734970368t^{11} + 81793284533694144552192t^{10} + 6621474159129065905712t^{9} + 403524443535398850192t^{8} + 17739909732234675328t^{7} + 506280495978197032t^{6} + 7455905890618631t^{5} + 23089053783897t^{4} - 633311608284t^{3} - 14333193676t^{2} - 28168339t - 9405 \right)D^{4} + t \left( 6t + 1 \right) \left( 10040729459780296442926817280t^{17} + 14006132353350654987994214400t^{16} + 9053185659854085317989232640t^{15} + 3616213424258184502076246016t^{14} + 1002072127934467048420532736t^{13} + 204209467654787367099607296t^{12} + 31539087345950709052364800t^{11} + 3743491744376534286730496t^{10} + 342827934697468135426464t^{9} + 24076198733643249159248t^{8} + 1264187573372336105904t^{7} + 46615183480661605128t^{6} + 1065511670908486542t^{5} + 12032106591289007t^{4} + 46120118611614t^{3} - 12728962734t^{2} - 54452488t + 113355 \right)D^{3} + 2 t ^{2} \left( 66555692419115107850257760256t^{17} + 99946915941798013238084911104t^{16} + 69853619395960092674073096192t^{15} + 30296646912971083556474929152t^{14} + 9156343523613471789873481728t^{13} + 2046281450921336301237021696t^{12} + 349099197727464735373390080t^{11} + 46193002484821816806747136t^{10} + 4768884758915184337049344t^{9} + 383178080305426134364608t^{8} + 23602676077605241701648t^{7} + 1072134875545528409024t^{6} + 33176174941453521032t^{5} + 604291727961122580t^{4} + 4979224133966575t^{3} + 16190175284020t^{2} + 78697217999t + 93615014 \right)D^{2} + 8 t ^{2} \left( 18073313027604533597268271104t^{17} + 26347478629890714721016684544t^{16} + 17883859214390336014458839040t^{15} + 7540912687121290737269858304t^{14} + 2218157015065087850288653824t^{13} + 482685819203765615010513408t^{12} + 80118238079642308217776896t^{11} + 10292921343680365256096256t^{10} + 1028424540708486452044704t^{9} + 79587945183889152733024t^{8} + 4680219443744454642752t^{7} + 199484501947236021984t^{6} + 5607183868411882946t^{5} + 87172464912282518t^{4} + 531163798061763t^{3} + 1273646574234t^{2} + 10407053693t + 14665530 \right)D + 144 t ^{2} \left( 409825692235930467058237440t^{17} + 585295735410061213691658240t^{16} + 389236903666898830860754944t^{15} + 160897514489392349465837568t^{14} + 46426150709940745057738752t^{13} + 9910546641879759297248256t^{12} + 1612136949368424152586496t^{11} + 202569805926754405173248t^{10} + 19738595726239456775616t^{9} + 1483256395855582195008t^{8} + 84026309389613550784t^{7} + 3397057252223447424t^{6} + 87962425671120008t^{5} + 1189587816163444t^{4} + 5431129287849t^{3} + 6440355288t^{2} + 68567345t + 106590 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ5P5

|λ| = 69.52307 λ ≈ 69.52307; multiplicity 1; Jordan blocks 1 λ ≈ -4.690793 - 24.99095i; multiplicity 1; Jordan blocks 1 λ ≈ -4.690793 + 24.99095i; multiplicity 1; Jordan blocks 1 λ ≈ -6.070741 - 0.1956947i; multiplicity 1; Jordan blocks 1 λ ≈ -6.070741 + 0.1956947i; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 4; Jordan blocks 1, 3 4
Sector dimension
9
Spectral radius
$69.5230676$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$69.52307$11
$-4.690793 - 24.99095\mathrm{i}$11
$-4.690793 + 24.99095\mathrm{i}$11
$-6.070741 - 0.1956947\mathrm{i}$11
$-6.070741 + 0.1956947\mathrm{i}$11
$-6$41, 3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 6\right)^{4} \left(\lambda^{5} - 48 \lambda^{4} - 699 \lambda^{3} - 47238 \lambda^{2} - 545972 \lambda - 1658312\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 6\right)^{3} \left(\lambda^{5} - 48 \lambda^{4} - 699 \lambda^{3} - 47238 \lambda^{2} - 545972 \lambda - 1658312\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 48 & 156 & 2916 & 1296 & 4716 & 84192 & 40464 & 581760 \\ 1 & -1 & 17 & 124 & 30 & 302 & 4434 & 2340 & 40320 \\ 1 & 1 & 9 & 192 & 126 & 314 & 7644 & 3978 & 53952 \\ 0 & 1 & 1 & 8 & -3 & 29 & 116 & 24 & 1584 \\ 0 & 1 & 0 & 17 & 0 & 23 & 302 & 114 & 1872 \\ 0 & 0 & 1 & 1 & 3 & 0 & 198 & 126 & 1728 \\ 0 & 0 & 0 & 1 & 0 & \frac{7}{3} & \frac{1}{3} & 1 & 64 \\ 0 & 0 & 0 & 1 & 1 & -\frac{4}{9} & \frac{257}{9} & \frac{23}{3} & \frac{368}{3} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{4} & \frac{3}{4} & 0\end{pmatrix}\notag\end{equation}$$