Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-105-28-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ1P1 × Gr(2, 5)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
2 copies of E^(w4) + E^(w2 + w4) + E^(w1 + w4)
Bundle
2 copies of O_3(1) + O_2(1) box O_3(1) + O_1(1) box O_3(1)
ZL62 label
1141.33mi
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 28
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 105
$\chi(X, \mathrm{T}_X)$ = -16

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 105
$\mathrm{c}_1^2\mathrm{c}_2$ = 114
$\mathrm{c}_2^2$ = 120
$\mathrm{c}_1\mathrm{c}_3$ = 34
$\mathrm{c}_4$ = 25
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 17 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 + 16 0 x0 + 16 x1
$\mathrm{HH}^{ 3 }$ = x1 + 20 0 0 x1 + 16 4
$\mathrm{HH}^{ 4 }$ = 28 0 0 0 0 28
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 23t + 57t^{2} + 23t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(623)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) No match

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( 5t + 1 \right) \left( 4t + 1 \right) ^{2} \left( 29t^{2} + 3t - 1 \right) \left( 1025t^{4} + 745t^{3} + 194t^{2} + 12t - 1 \right) \left( 1850754888800t^{11} + 3130237761600t^{10} + 2463689755280t^{9} + 1198679271008t^{8} + 398477936016t^{7} + 92651047528t^{6} + 14607602318t^{5} + 1427453744t^{4} + 69074320t^{3} + 286215t^{2} - 62210t + 231 \right)D^{6} + 2 \left( 4t + 1 \right) \left( 11552874704611800000t^{19} + 32168695821907450000t^{18} + 42211923178308158000t^{17} + 34673991711578735400t^{16} + 19939531221569146200t^{15} + 8477313981627082520t^{14} + 2730270909538517252t^{13} + 667347722406571815t^{12} + 121086911183895307t^{11} + 15318339275920468t^{10} + 1112026248852539t^{9} - 4128604703705t^{8} - 10746212298860t^{7} - 1217081154431t^{6} - 74808592218t^{5} - 3407620725t^{4} - 124402768t^{3} - 853166t^{2} + 92853t - 231 \right)D^{5} + \left( 770191646974120000000t^{20} + 2221825398527328000000t^{19} + 3035024823485133716000t^{18} + 2612111174656958008800t^{17} + 1586947255532619229200t^{16} + 720447298223126213200t^{15} + 251426701273156342224t^{14} + 68098722053603981352t^{13} + 14235834029458217414t^{12} + 2249346114077659046t^{11} + 257329356895677036t^{10} + 19469531490889979t^{9} + 746500652855347t^{8} - 6880886522963t^{7} - 1069608539028t^{6} + 126447084694t^{5} + 15998069173t^{4} + 601589722t^{3} + 4189825t^{2} - 107502t + 231 \right)D^{4} + 2 t \left( 1617402458645652000000t^{19} + 4475584649690836000000t^{18} + 5877648785767988150000t^{17} + 4877510572353577116400t^{16} + 2865737950281511825400t^{15} + 1261113994327335124680t^{14} + 427066130032022834136t^{13} + 112208167192942567760t^{12} + 22734792930956937829t^{11} + 3483230324418073789t^{10} + 388732777302425276t^{9} + 29399036179220959t^{8} + 1258741808467991t^{7} + 8313551278499t^{6} - 1646947797048t^{5} - 37672300332t^{4} + 1324836506t^{3} + 15178517t^{2} - 1252113t - 374 \right)D^{3} + 4 t ^{2} \left( 1786844620979958400000t^{18} + 4783649984511123000000t^{17} + 6089285027658819708000t^{16} + 4908962504003995708000t^{15} + 2807793452746436672200t^{14} + 1204310905820192524200t^{13} + 397370486987347567896t^{12} + 101525202377812868328t^{11} + 19936952734757119237t^{10} + 2949376047266064721t^{9} + 316896220790151183t^{8} + 23097844625718759t^{7} + 971131536706613t^{6} + 9406828090218t^{5} - 968124643851t^{4} - 29948817992t^{3} + 22171772t^{2} - 11728028t - 386631 \right)D^{2} + 8 t ^{2} \left( 970441475187391200000t^{18} + 2534016606801334600000t^{17} + 3150423245822229502000t^{16} + 2484502382469236761200t^{15} + 1392091949794893613600t^{14} + 585224121654102670800t^{13} + 189087809865797385312t^{12} + 47186983813066557584t^{11} + 9014789388575301602t^{10} + 1290775651717087158t^{9} + 133409564410092215t^{8} + 9279352371321479t^{7} + 366429448625592t^{6} + 2727815928075t^{5} - 381398510053t^{4} - 12275864289t^{3} - 105868749t^{2} - 6197433t - 64218 \right)D + 32 t ^{2} \left( 99024640325244000000t^{18} + 254161047836487000000t^{17} + 310844268171967422000t^{16} + 241388955072135414000t^{15} + 133293658763565859600t^{14} + 55231312670386010080t^{13} + 17571138120618580424t^{12} + 4307165279120561496t^{11} + 805278408664676700t^{10} + 112277313132218780t^{9} + 11224053107011852t^{8} + 746897626397260t^{7} + 27382426985518t^{6} + 89980701290t^{5} - 33586667033t^{4} - 1071575567t^{3} - 13539756t^{2} - 493603t - 2541 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × Gr(2, 5)

|λ| = 22.29356 λ ≈ 22.29356; multiplicity 1; Jordan blocks 1 λ ≈ 7.09017; multiplicity 2; Jordan blocks 1, 1 λ ≈ -5.11322; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 1; Jordan blocks 1 λ ≈ -4.09017; multiplicity 2; Jordan blocks 1, 1 λ ≈ -4; multiplicity 4; Jordan blocks 1, 1, 1, 1 λ ≈ -2.59017 - 1.510923i; multiplicity 1; Jordan blocks 1 λ ≈ -2.59017 + 1.510923i; multiplicity 1; Jordan blocks 1 2 2 4
Sector dimension
13
Spectral radius
$22.2935596$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$22.29356$11
$7.09017$21, 1
$-5.11322$11
$-5$11
$-4.09017$21, 1
$-4$41, 1, 1, 1
$-2.59017 - 1.510923\mathrm{i}$11
$-2.59017 + 1.510923\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{4} \left(\lambda + 5\right) \left(\lambda^{2} - 3 \lambda - 29\right)^{2} \left(\lambda^{4} - 12 \lambda^{3} - 194 \lambda^{2} - 745 \lambda - 1025\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right) \left(\lambda + 5\right) \left(\lambda^{2} - 3 \lambda - 29\right) \left(\lambda^{4} - 12 \lambda^{3} - 194 \lambda^{2} - 745 \lambda - 1025\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 6 & 6 & 32 & 9 & 69 & 69 & 93 & 141 & 160 & 208 & 208 & 490 \\ 1 & -1 & 0 & 3 & 6 & 26 & 6 & 14 & 22 & 60 & 66 & 36 & 144 \\ 1 & 0 & -1 & 3 & 6 & 6 & 26 & 14 & 22 & 60 & 36 & 66 & 144 \\ 1 & 1 & 1 & 2 & 0 & 10 & 10 & 16 & 24 & 15 & 30 & 30 & 64 \\ 0 & 1 & 1 & 0 & -2 & 3 & 3 & 1 & 2 & 20 & 12 & 12 & 42 \\ 0 & 1 & 0 & 1 & 1 & 1 & 0 & 2 & 3 & 10 & 12 & 4 & 24 \\ 0 & 0 & 1 & 1 & 1 & 0 & 1 & 2 & 3 & 10 & 4 & 12 & 24 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & -3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & 2 & 0 & 0 & 4 & 4 & 6 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & -\frac{2}{5} & -\frac{3}{5} & 0 & \frac{8}{5} & \frac{8}{5} & 8 \\ 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 0 & 2 & 3 & \frac{5}{2} & 0 & 1 & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 2 & 3 & \frac{5}{2} & 1 & 0 & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$