Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-30-13-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ6P6
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(2w2) + E^(3w2) + E^(w1 + w2)
Bundle
O_2(2) + O_2(3) + O_1(1) box O_2(1)
ZL62 label
1600.8Fw0
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;7) \\ \det\mathcal{E} &\cong \mathcal{O}(1;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 13
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 30
$\chi(X, \mathrm{T}_X)$ = -65

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 30
$\mathrm{c}_1^2\mathrm{c}_2$ = 84
$\mathrm{c}_2^2$ = 192
$\mathrm{c}_1\mathrm{c}_3$ = -48
$\mathrm{c}_4$ = 114
Hodge diamond
1
00
0 2 0
0 0 0 0
0 9 90 9 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 + 65 0 x0 + 65 x2
$\mathrm{HH}^{ 3 }$ = x1 + 1 0 0 x1 1
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 55 0 0 x1 - x2 + 42 0 13
$\mathrm{HH}^{ 5 }$ = x3 0 0 x3 0
$\mathrm{HH}^{ 6 }$ = x3 - 6 0 x3 - 6 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 8t + 12t^{2} + 8t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(737)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(728)

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( 12t + 1 \right) ^{3} \left( 13t + 1 \right) ^{3} \left( 8784t^{2} - 192t + 1 \right) \left( 516287164416t^{6} + 122902373376t^{5} + 8657933824t^{4} + 62609088t^{3} - 11295520t^{2} - 196400t + 125 \right)D^{6} + \left( 12t + 1 \right) ^{2} \left( 13t + 1 \right) ^{2} \left( 14856877697505951744t^{10} + 4625866685131849728t^{9} + 500456790690250752t^{8} + 18170774508613632t^{7} - 340833783736320t^{6} - 33316329037824t^{5} - 424083009600t^{4} - 159120560t^{3} + 45854980t^{2} + 567575t - 250 \right)D^{5} + \left( 12t + 1 \right) \left( 13t + 1 \right) \left( 19313941006757737267200t^{12} + 8103141403492787159040t^{11} + 1341797499450664943616t^{10} + 106384828034447474688t^{9} + 3374326685180301312t^{8} - 57375710857095168t^{7} - 7263354496432384t^{6} - 185191115728000t^{5} - 1494556139936t^{4} + 2405737976t^{3} - 33066555t^{2} - 363800t + 125 \right)D^{4} + t \left( 12654494147627669457469440t^{13} + 6819424094380970952622080t^{12} + 1547188253351520341852160t^{11} + 188745182264519941423104t^{10} + 12778572000016713793536t^{9} + 390345354843445112832t^{8} - 5591173697770171392t^{7} - 910233385951318272t^{6} - 31554378809024256t^{5} - 487609942475360t^{4} - 2883517337836t^{3} + 477263961t^{2} - 34581280t - 3375 \right)D^{3} + 2 t ^{2} \left( 13980203058331520543490048t^{12} + 7096245010438838646472704t^{11} + 1500527341477023940214784t^{10} + 167667069372240844652544t^{9} + 9989190999801290465280t^{8} + 218900399851075811328t^{7} - 9298012317568700928t^{6} - 785929369206536832t^{5} - 22480864420992032t^{4} - 294056435478960t^{3} - 1402361834060t^{2} + 734248715t - 9459950 \right)D^{2} + 24 t ^{2} \left( 1265449414762766945746944t^{12} + 613305558826743700389888t^{11} + 122635086496758238740480t^{10} + 12746424072069014175744t^{9} + 677487843401805926400t^{8} + 9625522733050840064t^{7} - 904541106345942784t^{6} - 56753318114018624t^{5} - 1403738168709040t^{4} - 15901817756088t^{3} - 61040810780t^{2} + 13978775t - 224875 \right)D + 192 t ^{2} \left( 64563745651161578864640t^{12} + 30290562759216465838080t^{11} + 5816126658512795369472t^{10} + 572270259201548894208t^{9} + 27679935077236666368t^{8} + 212565796711799808t^{7} - 45931194519860736t^{6} - 2421008308136736t^{5} - 53672449889560t^{4} - 539880752880t^{3} - 1695212800t^{2} - 846125t - 4375 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ6P6

|λ| = 116.7846 λ ≈ 116.7846; multiplicity 1; Jordan blocks 1 λ ≈ 75.21539; multiplicity 1; Jordan blocks 1 λ ≈ -13; multiplicity 3; Jordan blocks 1, 2 λ ≈ -12; multiplicity 3; Jordan blocks 3 3 3
Sector dimension
8
Spectral radius
$116.78461$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$116.7846$11
$75.21539$11
$-13$31, 2
$-12$33

Characteristic polynomial

$$\begin{equation}\left(\lambda + 12\right)^{3} \left(\lambda + 13\right)^{3} \left(\lambda^{2} - 192 \lambda + 8784\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 12\right)^{3} \left(\lambda + 13\right)^{2} \left(\lambda^{2} - 192 \lambda + 8784\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 24 & 816 & 2376 & 46872 & 175680 & 1010304 & 4034880 \\ 1 & -1 & 12 & 792 & 1368 & 44496 & 65232 & 834624 \\ 1 & 1 & 30 & 84 & 2424 & 7020 & 51516 & 175680 \\ 0 & 1 & 1 & 29 & 42 & 2340 & 2916 & 44496 \\ 0 & 0 & 1 & 1 & 30 & 84 & 876 & 2376 \\ 0 & 0 & 0 & 1 & 1 & 29 & 42 & 792 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 24 \\ 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0\end{pmatrix}\notag\end{equation}$$