Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-196-46-a

·

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

Defining data
Ambient
ℙ3P3 × ℙ6P6
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_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9ω1ω2ω3ω4ω5ω6ω7ω8ω9
Bundle
E^(2w4) + E^(w1 + w4) + E^(w3 + w4)
Bundle
O_2(2) + O_1(1) box O_2(1) + Q_1 box O_2(1)
ZL62 label
30600.CwFw2
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;7) \\ \det\mathcal{E} &\cong \mathcal{O}(2;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 46
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 196
$\chi(X, \mathrm{T}_X)$ = -11

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 196
$\mathrm{c}_1^2\mathrm{c}_2$ = 148
$\mathrm{c}_2^2$ = 100
$\mathrm{c}_1\mathrm{c}_3$ = 40
$\mathrm{c}_4$ = 16
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 10 0 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 + x1 + 11 0 x0 + 11 x1
$\mathrm{HH}^{ 3 }$ = x1 + 42 0 0 x1 + 3 39
$\mathrm{HH}^{ 4 }$ = 46 0 0 0 0 46
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 41t + 112t^{2} + 41t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(123)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(168)

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) ^{2} \left( 8t^{2} + 4t + 1 \right) \left( 7744t^{5} + 4224t^{4} + 488t^{3} - 12t^{2} + 6t - 1 \right) \left( 13035137662976t^{13} + 40975469248512t^{12} + 56978204672000t^{11} + 45889659568128t^{10} + 24005681999872t^{9} + 8714480585728t^{8} + 2264265790976t^{7} + 417555881856t^{6} + 52279225280t^{5} + 4240614272t^{4} + 228271936t^{3} + 8025345t^{2} + 24729t + 539 \right)D^{6} + \left( 2t + 1 \right) \left( 33917219636860944384t^{21} + 150897560018636242944t^{20} + 310727861934508474368t^{19} + 392219287298550792192t^{18} + 340263247553845264384t^{17} + 215965235237722521600t^{16} + 104200913839628484608t^{15} + 39124867204832296960t^{14} + 11570730515259981824t^{13} + 2701497011636502528t^{12} + 495221579688304640t^{11} + 70632373583380480t^{10} + 7822872144078848t^{9} + 694811747808256t^{8} + 55346004116992t^{7} + 4588513277184t^{6} + 376047931120t^{5} + 24169257120t^{4} + 1081232432t^{3} + 31938044t^{2} + 69875t + 1078 \right)D^{5} + \left( 565286993947682406400t^{22} + 2689103142411766333440t^{21} + 5929996409170372329472t^{20} + 8032878369727473254400t^{19} + 7495526958326513401856t^{18} + 5130100699512600264704t^{17} + 2678100839279189557248t^{16} + 1093188520728220663808t^{15} + 353881239671978983424t^{14} + 91300362567900463104t^{13} + 18720634183923056640t^{12} + 3023572807295852544t^{11} + 379399743031280640t^{10} + 36227619366127360t^{9} + 2512064021033088t^{8} + 105255363178912t^{7} - 611438347968t^{6} - 455086298860t^{5} - 30682986276t^{4} - 1066563854t^{3} - 28160453t^{2} - 46796t - 539 \right)D^{4} + 2 t \left( 1187102687290133053440t^{21} + 5468092113524789084160t^{20} + 11646884043223214325760t^{19} + 15194458104656646111232t^{18} + 13622946271257365053440t^{17} + 8947048922179275063296t^{16} + 4479501269443126493184t^{15} + 1752727711585068384256t^{14} + 543063266099222740992t^{13} + 133705416330758176768t^{12} + 26060475018016088064t^{11} + 3990455855708045312t^{10} + 476276551092393984t^{9} + 44056099658869120t^{8} + 3131914994856448t^{7} + 165438772169408t^{6} + 5839230462160t^{5} + 105737445186t^{4} + 499858572t^{3} - 258314t^{2} + 114321t + 286 \right)D^{3} + 4 t ^{2} \left( 1311465825958623182848t^{20} + 5889621543592453472256t^{19} + 12203818458627377725440t^{18} + 15448817897643016454144t^{17} + 13414374274159781871616t^{16} + 8523983197296161980416t^{15} + 4127467774104137498624t^{14} + 1560892801459322290176t^{13} + 466568824835348332544t^{12} + 110446286598442311680t^{11} + 20608092764972128256t^{10} + 3009391527837127168t^{9} + 341984122142459904t^{8} + 30130308866388032t^{7} + 2033831853628512t^{6} + 100189251749808t^{5} + 3174845486624t^{4} + 52070512288t^{3} + 557652185t^{2} + 9198524t + 50813 \right)D^{2} + 16 t ^{2} \left( 356130806187039916032t^{20} + 1569224713672241184768t^{19} + 3184491085998818590720t^{18} + 3939545285098504978432t^{17} + 3337572234951217119232t^{16} + 2067575419477614395392t^{15} + 975637220755240091648t^{14} + 359267326311085473792t^{13} + 104362231341272793088t^{12} + 23928726506287835136t^{11} + 4307647185541209600t^{10} + 605006983704272896t^{9} + 66050250747005056t^{8} + 5587980494303856t^{7} + 359707406785536t^{6} + 16457076437192t^{5} + 455717727297t^{4} + 6080919321t^{3} + 84081341t^{2} + 1455841t + 5929 \right)D + 16 t ^{2} \left( 145359512729404047360t^{20} + 632196335564740362240t^{19} + 1264509144370520260608t^{18} + 1539245138221260079104t^{17} + 1281482524495448113152t^{16} + 779577090717912662016t^{15} + 361086355362261565440t^{14} + 130406702518683369472t^{13} + 37083293575753351168t^{12} + 8298989088459141120t^{11} + 1453304508782195712t^{10} + 198054110401334784t^{9} + 20962497604606976t^{8} + 1718203963769248t^{7} + 106400304941920t^{6} + 4568823090616t^{5} + 111796572474t^{4} + 1180889988t^{3} + 18938257t^{2} + 273306t + 1078 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × ℙ6P6

|λ| = 11.65312 λ ≈ 11.65312; multiplicity 1; Jordan blocks 1 λ ≈ 0.17132 - 8.395009i; multiplicity 1; Jordan blocks 1 λ ≈ 0.17132 + 8.395009i; multiplicity 1; Jordan blocks 1 λ ≈ -2.997882 - 0.6618961i; multiplicity 1; Jordan blocks 1 λ ≈ -2.997882 + 0.6618961i; multiplicity 1; Jordan blocks 1 λ ≈ -2 - 2i; multiplicity 1; Jordan blocks 1 λ ≈ -2 + 2i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 2 2
Sector dimension
9
Spectral radius
$11.6531233$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$11.65312$11
$0.17132 - 8.395009\mathrm{i}$11
$0.17132 + 8.395009\mathrm{i}$11
$-2.997882 - 0.6618961\mathrm{i}$11
$-2.997882 + 0.6618961\mathrm{i}$11
$-2 - 2\mathrm{i}$11
$-2 + 2\mathrm{i}$11
$-2$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda^{2} + 4 \lambda + 8\right) \left(\lambda^{5} - 6 \lambda^{4} + 12 \lambda^{3} - 488 \lambda^{2} - 4224 \lambda - 7744\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda^{2} + 4 \lambda + 8\right) \left(\lambda^{5} - 6 \lambda^{4} + 12 \lambda^{3} - 488 \lambda^{2} - 4224 \lambda - 7744\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 4 & 12 & 12 & 12 & 96 & 80 & 160 \\ 2 & 0 & 4 & 4 & -2 & 8 & 0 & 0 & 32 \\ 1 & 0 & 0 & 0 & 2 & 0 & 24 & 24 & 64 \\ 0 & 1 & 2 & 0 & 0 & 4 & 0 & -2 & 0 \\ 0 & 2 & 0 & 4 & 0 & 4 & 8 & 4 & 0 \\ 0 & 0 & 1 & 0 & 0 & -2 & 0 & 2 & 12 \\ 0 & 0 & 0 & 1 & -1 & 3 & -4 & -2 & -4 \\ 0 & 0 & 0 & 2 & 3 & 0 & 8 & 4 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 2 & 0\end{pmatrix}\notag\end{equation}$$