Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-156-38-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ2P2 × ℙ6P6
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)node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9node 10: fundamental weight ω_10ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10
Bundle
E^(2w5) + E^(w4 + w5) + E^(w1 + w3) + E^(w2 + w5)
Bundle
O_3(2) + Q_2 box O_3(1) + O_1(1) box O_2(1) + Q_1 box O_3(1)
ZL62 label
2020600.EwOwBw1
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3;7) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 38
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 156
$\chi(X, \mathrm{T}_X)$ = -10

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 156
$\mathrm{c}_1^2\mathrm{c}_2$ = 132
$\mathrm{c}_2^2$ = 108
$\mathrm{c}_1\mathrm{c}_3$ = 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 + 10 0 x0 + 10 x1
$\mathrm{HH}^{ 3 }$ = x1 + 29 0 0 x1 + 9 20
$\mathrm{HH}^{ 4 }$ = 38 0 0 0 0 38
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 33t + 88t^{2} + 33t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(476)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(528)

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( 3t + 1 \right) \left( 2t + 1 \right) ^{2} \left( 17t^{2} - 2t + 1 \right) \left( 648t^{3} + 148t^{2} + 2t - 1 \right) \left( 3075425386164t^{9} + 2727056525040t^{8} + 1085812097200t^{7} + 264431854187t^{6} + 34300529893t^{5} + 1156803188t^{4} + 25968161t^{3} + 5790720t^{2} - 159338t + 273 \right)D^{6} + 3 \left( 2t + 1 \right) \left( 1422913214267270208t^{16} + 2273770078244648064t^{15} + 1649852318969630496t^{14} + 741366578570838784t^{13} + 230620250947016188t^{12} + 50981701470798636t^{11} + 8186082010485732t^{10} + 1027574530868368t^{9} + 103835941182759t^{8} + 8441762778343t^{7} + 916789815766t^{6} + 96741519486t^{5} + 2804067629t^{4} - 8978841t^{3} + 8276688t^{2} - 160248t + 182 \right)D^{5} + \left( 71145660713363510400t^{17} + 136008997189510397760t^{16} + 116764991635880744784t^{15} + 60785728900715056704t^{14} + 21514804794873325832t^{13} + 5348360333327628844t^{12} + 923976534812736436t^{11} + 109621063827158726t^{10} + 8939747625823237t^{9} + 431766636684946t^{8} - 1995680460786t^{7} - 2228289583551t^{6} - 291284523648t^{5} - 12778957153t^{4} - 253907726t^{3} - 23861171t^{2} + 283599t - 273 \right)D^{4} + t \left( 298811774996126743680t^{16} + 527505075833911260480t^{15} + 422264752714508519520t^{14} + 206643585093135059616t^{13} + 68657268970624430520t^{12} + 15724872365999529628t^{11} + 2406804691575958820t^{10} + 238693806867338808t^{9} + 15882350621468010t^{8} + 847487942514553t^{7} + 56592228745103t^{6} + 4377257096542t^{5} + 153220852840t^{4} - 9225621965t^{3} - 197566699t^{2} + 1960314t + 406 \right)D^{3} + 2 t ^{2} \left( 330115865710006688256t^{15} + 545802957853365045888t^{14} + 412830714794871736152t^{13} + 192375667383559853664t^{12} + 60771544098116452048t^{11} + 13011040557449109768t^{10} + 1802514575790087956t^{9} + 155827079747905362t^{8} + 9326019995001182t^{7} + 587125836540385t^{6} + 46961913421639t^{5} + 2360392990537t^{4} + 22942373503t^{3} - 5661448570t^{2} - 20930564t + 536250 \right)D^{2} + 8 t ^{2} \left( 89643532498838023104t^{15} + 140857830333455942304t^{14} + 101966547347857476324t^{13} + 45767948198638623888t^{12} + 13893314287687455492t^{11} + 2813425539761575102t^{10} + 358922820350240180t^{9} + 28122284101404809t^{8} + 1689697324868871t^{7} + 130999969497733t^{6} + 9896789335609t^{5} + 311501763268t^{4} - 5512095380t^{3} - 791969270t^{2} + 2786564t + 37674 \right)D + 16 t ^{2} \left( 18294598469150616960t^{15} + 27732286516057047360t^{14} + 19457224043052139416t^{13} + 8502853377168155952t^{12} + 2505997328218928352t^{11} + 486004014931014684t^{10} + 58108196692250164t^{9} + 4261467764407174t^{8} + 271480254822622t^{7} + 24144908978993t^{6} + 1626020697527t^{5} + 33649238537t^{4} - 1655912917t^{3} - 84682402t^{2} + 679616t + 2730 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2 × ℙ6P6

|λ| = 14.87654 λ ≈ 14.87654; multiplicity 1; Jordan blocks 1 λ ≈ -6.438272 - 1.451602i; multiplicity 1; Jordan blocks 1 λ ≈ -6.438272 + 1.451602i; multiplicity 1; Jordan blocks 1 λ ≈ 1 - 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 1 + 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -3; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2; multiplicity 3; Jordan blocks 1, 2 2 2 2 3
Sector dimension
12
Spectral radius
$14.876545$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$14.87654$11
$-6.438272 - 1.451602\mathrm{i}$11
$-6.438272 + 1.451602\mathrm{i}$11
$1 - 4\mathrm{i}$21, 1
$1 + 4\mathrm{i}$21, 1
$-3$21, 1
$-2$31, 2

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda + 3\right)^{2} \left(\lambda^{2} - 2 \lambda + 17\right)^{2} \left(\lambda^{3} - 2 \lambda^{2} - 148 \lambda - 648\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 3\right) \left(\lambda^{2} - 2 \lambda + 17\right) \left(\lambda^{3} - 2 \lambda^{2} - 148 \lambda - 648\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 12 & 12 & 24 & 6 & 24 & 96 & 32 & 112 & 240 \\ 1 & -1 & 0 & 2 & 4 & 8 & 0 & 8 & 12 & 0 & 12 & 32 \\ 1 & 0 & -1 & 2 & 4 & 8 & 4 & 8 & 12 & 0 & 24 & 32 \\ 1 & 1 & 1 & 0 & 0 & 4 & 0 & 4 & 24 & 12 & 24 & 64 \\ 0 & 1 & 2 & -1 & -2 & 1 & -1 & 3 & 12 & 4 & 12 & 12 \\ 0 & 1 & 0 & 2 & 1 & 0 & 0 & 1 & 4 & 0 & 4 & 12 \\ 0 & 1 & -1 & 0 & 0 & 2 & 0 & -2 & 0 & 0 & 4 & 0 \\ 0 & 0 & 1 & 2 & 1 & 1 & 1 & 0 & 4 & 0 & 8 & 12 \\ 0 & 0 & 0 & 0 & 1 & 0 & -1 & 6 & 1 & 1 & 2 & 8 \\ 0 & 0 & 0 & 0 & 2 & 1 & 2 & -3 & 2 & -2 & 4 & 8 \\ 0 & 0 & 0 & 0 & 0 & 2 & 1 & -2 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{1}{2} & \frac{5}{2} & 0\end{pmatrix}\notag\end{equation}$$