Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-432-90-c

·

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

Defining data
Ambient
ℙ2P2 × ℙ1P1 × ℙ1P1
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4 (marked)ω1ω2ω3ω4
Bundle
The bundle is zero: this presentation is the ambient variety itself.
ZL62 label
1120
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;2;2) \\ \det\mathcal{E} &\cong \mathcal{O}(0;0;0) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 90
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 432
$\chi(X, \mathrm{T}_X)$ = 14

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 432
$\mathrm{c}_1^2\mathrm{c}_2$ = 204
$\mathrm{c}_2^2$ = 96
$\mathrm{c}_1\mathrm{c}_3$ = 60
$\mathrm{c}_4$ = 12
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 4 0 0
0 0 0 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 14 0 14
$\mathrm{HH}^{ 2 }$ = 67 0 0 67
$\mathrm{HH}^{ 3 }$ = 132 0 0 0 132
$\mathrm{HH}^{ 4 }$ = 90 0 0 0 0 90
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 85t + 260t^{2} + 85t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(140)
Coates et al. (2020) P2 x P1 x P1
Coates–Kasprzyk–Prince (2015) CKP(195)

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( t + 1 \right) \left( 3t - 1 \right) \left( 7t - 1 \right) \left( 9t^{2} + 3t + 1 \right) \left( 13t^{2} - 5t + 1 \right) \left( 37t^{2} + 11t + 1 \right) \left( 1487913984t^{11} + 3937498128t^{10} - 3567972672t^{9} + 1794239488t^{8} + 331814610t^{7} + 692173216t^{6} - 54038826t^{5} - 98326t^{4} + 2457216t^{3} + 144891t^{2} - 11664t + 768 \right)D^{6} + 2 \left( 1420280109900288t^{20} + 4822600654680048t^{19} - 1450639695356640t^{18} - 999126937013760t^{17} + 2145229092047565t^{16} + 786931562241504t^{15} + 406884811946565t^{14} - 189752644819917t^{13} + 5795490865328t^{12} - 1146273697800t^{11} + 1693083815807t^{10} - 236299281264t^{9} + 9067489536t^{8} + 4092133473t^{7} - 3303357920t^{6} + 91761660t^{5} + 284224t^{4} - 8009280t^{3} - 508401t^{2} + 29160t - 1536 \right)D^{5} + \left( 23671335165004800t^{20} + 79986233099956320t^{19} - 33713410776662592t^{18} - 4632061281693696t^{17} + 32746807675746342t^{16} + 17034662536605504t^{15} + 6795645124599046t^{14} - 2633851873944288t^{13} + 108169884683136t^{12} + 27831862647882t^{11} + 1706080850596t^{10} - 1441434700800t^{9} + 101616670188t^{8} + 28763731152t^{7} + 22964671680t^{6} - 1207012833t^{5} + 21823318t^{4} + 28810080t^{3} + 1059795t^{2} - 58320t + 3072 \right)D^{4} + t ^{2} \left( 99419607693020160t^{18} + 334653712242269760t^{17} - 176429267427654720t^{16} + 23799498091269120t^{15} + 126052046344592622t^{14} + 86527992482448768t^{13} + 28079834098173350t^{12} - 9439140526351830t^{11} + 586211052675552t^{10} + 226989631485342t^{9} - 28235180493230t^{8} - 3315527630496t^{7} - 671354814797t^{6} + 22453889310t^{5} + 16347588352t^{4} - 723555504t^{3} + 125992408t^{2} - 5920128t - 159744 \right)D^{3} + t ^{2} \left( 219669990331244544t^{18} + 737247222735162192t^{17} - 452880125064201600t^{16} + 130249810050416640t^{15} + 257326080024316752t^{14} + 219264678020189088t^{13} + 59212648153430940t^{12} - 17794564739772192t^{11} + 1441931119172896t^{10} + 656187469391385t^{9} - 74693059866408t^{8} - 8042167919808t^{7} - 1456580654064t^{6} + 12990946104t^{5} + 24255737312t^{4} - 1515674754t^{3} + 125027520t^{2} - 8107776t - 49152 \right)D^{2} + 2 t ^{3} \left( 119303529231624192t^{17} + 399534621180990192t^{16} - 272207550673589760t^{15} + 102920025759436800t^{14} + 130844527269972228t^{13} + 131016195398569824t^{12} + 30200228009833880t^{11} - 8303237114446620t^{10} + 806110804534576t^{9} + 414036349260399t^{8} - 37704895553676t^{7} - 5048049612096t^{6} - 578178991194t^{5} - 1475136144t^{4} + 8450734128t^{3} - 786163968t^{2} + 26921976t - 2457216 \right)D + 24 t ^{3} \left( 4057943171143680t^{17} + 13569692328704880t^{16} - 9878019165902592t^{15} + 4258109326593024t^{14} + 4239099913207644t^{13} + 4740963072022656t^{12} + 968538688218608t^{11} - 248852872180548t^{10} + 27445278979192t^{9} + 15388356362895t^{8} - 1136620801668t^{7} - 189742972464t^{6} - 12910892454t^{5} - 202120326t^{4} + 187618200t^{3} - 24700032t^{2} + 274452t - 46656 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ1P1 × ℙ1P1

|λ| = 7 λ ≈ 7; multiplicity 1; Jordan blocks 1 λ ≈ -5.5 - 2.598076i; multiplicity 1; Jordan blocks 1 λ ≈ -5.5 + 2.598076i; multiplicity 1; Jordan blocks 1 λ ≈ 2.5 - 2.598076i; multiplicity 1; Jordan blocks 1 λ ≈ 2.5 + 2.598076i; multiplicity 1; Jordan blocks 1 λ ≈ -1.5 - 2.598076i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1.5 + 2.598076i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 3; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1; multiplicity 1; Jordan blocks 1 2 2 2
Sector dimension
12
Spectral radius
$7$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$7$11
$-5.5 - 2.598076\mathrm{i}$11
$-5.5 + 2.598076\mathrm{i}$11
$2.5 - 2.598076\mathrm{i}$11
$2.5 + 2.598076\mathrm{i}$11
$-1.5 - 2.598076\mathrm{i}$21, 1
$-1.5 + 2.598076\mathrm{i}$21, 1
$3$21, 1
$-1$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda - 3\right)^{2} \left(\lambda - 7\right) \left(\lambda^{2} + 3 \lambda + 9\right)^{2} \left(\lambda^{2} + 11 \lambda + 37\right) \left(\lambda^{2} - 5 \lambda + 13\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda - 3\right) \left(\lambda - 7\right) \left(\lambda^{2} + 3 \lambda + 9\right) \left(\lambda^{2} + 11 \lambda + 37\right) \left(\lambda^{2} - 5 \lambda + 13\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 2 & 2 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 \\ 3 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 3 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 3 & 0 \\ 0 & 2 & 3 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 3 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 \\ 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 & 2 & 2 & 0 & 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 0 & 2 & 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 3 & 2 & 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$