Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-70-21-a

·

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

Defining data
Ambient
Fl(1, 5, 6)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)ω1ω2ω3ω4ω5
Bundle
2 copies of E^(w1) + 2 copies of E^(w1 + w5) + E^(2w5)
Bundle
2 copies of O(1,0) + 2 copies of O(1,1) + O(0,2)
ZL62 label
5G.900cc
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5,5) \\ \det\mathcal{E} &\cong \mathcal{O}(4,4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 21
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 70
$\chi(X, \mathrm{T}_X)$ = -33

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 70
$\mathrm{c}_1^2\mathrm{c}_2$ = 100
$\mathrm{c}_2^2$ = 146
$\mathrm{c}_1\mathrm{c}_3$ = 4
$\mathrm{c}_4$ = 52
Hodge diamond
1
00
0 2 0
0 0 0 0
0 3 40 3 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 + 33 0 x0 + 33 x2
$\mathrm{HH}^{ 3 }$ = x1 + 3 0 0 x1 3
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 21 0 0 x1 - x2 0 21
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 16t + 36t^{2} + 16t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(659)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(666)

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) \left( 3t + 1 \right) ^{4} \left( 92t^{2} + 28t - 1 \right) \left( 132t^{2} + 20t + 1 \right) \left( 94731586560t^{10} + 218587548672t^{9} + 220085081856t^{8} + 125943846912t^{7} + 45159491712t^{6} + 10628881536t^{5} + 1666332064t^{4} + 166039680t^{3} + 8345260t^{2} + 37740t + 175 \right)D^{6} + \left( 3t + 1 \right) ^{3} \left( 144952968785264640t^{16} + 480968029608984576t^{15} + 723743652027654144t^{14} + 652054764009074688t^{13} + 391887726597863424t^{12} + 166024693230492672t^{11} + 51150463345583616t^{10} + 11644882353574656t^{9} + 1960101822254720t^{8} + 238692744517440t^{7} + 19876468491040t^{6} + 1017598646160t^{5} + 28324065384t^{4} + 732384660t^{3} + 33085330t^{2} + 109545t + 350 \right)D^{5} + \left( 3t + 1 \right) ^{2} \left( 3623824219631616000t^{17} + 12424637565102735360t^{16} + 19447351625255362560t^{15} + 18362021549294481408t^{14} + 11665709491453876224t^{13} + 5276603230754448384t^{12} + 1755502634683832832t^{11} + 437334699576423168t^{10} + 81918784870901376t^{9} + 11376502015623104t^{8} + 1124094181599840t^{7} + 72635455270864t^{6} + 2593128933960t^{5} + 32243579916t^{4} - 392445090t^{3} - 30638255t^{2} - 102930t - 175 \right)D^{4} + t \left( 3t + 1 \right) \left( 45660185167358361600t^{17} + 163776199684256317440t^{16} + 269836290919649157120t^{15} + 270066581616223309824t^{14} + 183319601099347224576t^{13} + 89377767325877031936t^{12} + 32366458676886391296t^{11} + 8875564914083745024t^{10} + 1856520720763670400t^{9} + 294190211417029056t^{8} + 34402710286521056t^{7} + 2812875143370480t^{6} + 145885647868344t^{5} + 4023052800828t^{4} + 44236084254t^{3} + 393827655t^{2} + 2322490t + 675 \right)D^{3} + 2 t ^{2} \left( 151330899411816284160t^{17} + 572973503146062446592t^{16} + 1001952615132679643136t^{15} + 1071041908303092068352t^{14} + 781979800743028039680t^{13} + 413267267244984367104t^{12} + 163601479949362343424t^{11} + 49516620106686725376t^{10} + 11570899818408609024t^{9} + 2084403751931802624t^{8} + 284915146163938272t^{7} + 28526186285013712t^{6} + 1963954737325440t^{5} + 83479606584024t^{4} + 1826369962194t^{3} + 18060576195t^{2} + 217099995t + 885080 \right)D^{2} + 8 t ^{2} \left( 41094166650622525440t^{17} + 151557987395111510016t^{16} + 257896132086273269760t^{15} + 267871791094151897088t^{14} + 189730935278267673600t^{13} + 97120109022701732352t^{12} + 37183210390397030400t^{11} + 10865997289566528768t^{10} + 2445499833721246656t^{9} + 422404162486741152t^{8} + 54918976785955680t^{7} + 5159718081824704t^{6} + 326140869038964t^{5} + 12295888182990t^{4} + 228726127260t^{3} + 2233700385t^{2} + 25517475t + 80150 \right)D + 96 t ^{2} \left( 1397760770429337600t^{17} + 5062335354829578240t^{16} + 8451596233503866880t^{15} + 8601790894154219520t^{14} + 5961452255316624384t^{13} + 2981688422066929152t^{12} + 1113884325834471936t^{11} + 317122007760969984t^{10} + 69371899099454016t^{9} + 11599357670807904t^{8} + 1449389741709504t^{7} + 129295842566752t^{6} + 7608958061820t^{5} + 258883329330t^{4} + 4215619530t^{3} + 43269585t^{2} + 386775t + 875 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × Gr(2, 4)

|λ| = 30.97056 λ ≈ 30.97056; multiplicity 1; Jordan blocks 1 λ ≈ -10 - 5.656854i; multiplicity 1; Jordan blocks 1 λ ≈ -10 + 5.656854i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 4; Jordan blocks 1, 3 λ ≈ -2.970563; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 1, 1 4 2
Sector dimension
10
Spectral radius
$30.9705627$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$30.97056$11
$-10 - 5.656854\mathrm{i}$11
$-10 + 5.656854\mathrm{i}$11
$-3$41, 3
$-2.970563$11
$-2$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 3\right)^{4} \left(\lambda^{2} + 20 \lambda + 132\right) \left(\lambda^{2} - 28 \lambda - 92\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right)^{3} \left(\lambda^{2} + 20 \lambda + 132\right) \left(\lambda^{2} - 28 \lambda - 92\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 28 & 32 & 372 & 192 & 192 & 192 & 1740 & 2392 & 6150 \\ 1 & -1 & 6 & 40 & 28 & 16 & 16 & 261 & 354 & 816 \\ 1 & 3 & 0 & 64 & 36 & 28 & 28 & 261 & 342 & 924 \\ 0 & 1 & 1 & -1 & 0 & 3 & 3 & 24 & 40 & 99 \\ 0 & 1 & 0 & 6 & -1 & 3 & 3 & 12 & 16 & 72 \\ 0 & 0 & 1 & 3 & 3 & -2 & 0 & 24 & 32 & 63 \\ 0 & 0 & 1 & 3 & 3 & 0 & -2 & 24 & 32 & 63 \\ 0 & 0 & 0 & 2 & -\frac{1}{3} & \frac{7}{3} & \frac{7}{3} & 0 & 6 & 26 \\ 0 & 0 & 0 & 1 & \frac{3}{2} & -\frac{1}{2} & -\frac{1}{2} & 3 & -1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{8}{3} & 0\end{pmatrix}\notag\end{equation}$$