Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-104-28-a

·

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

Defining data
Ambient
ℙ4P4 × ℙ7P7
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)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_10node 11: fundamental weight ω_11ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10ω11
Bundle
E^(2w5) + E^(w1 + w5) + E^(w4 + w5) + E^(2w1)
Bundle
O_2(2) + O_1(1) box O_2(1) + Q_1 box O_2(1) + O_1(2)
ZL62 label
40700.FwRw3B
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;8) \\ \det\mathcal{E} &\cong \mathcal{O}(4;7) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 28
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 104
$\chi(X, \mathrm{T}_X)$ = -27

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 104
$\mathrm{c}_1^2\mathrm{c}_2$ = 116
$\mathrm{c}_2^2$ = 128
$\mathrm{c}_1\mathrm{c}_3$ = 16
$\mathrm{c}_4$ = 40
Hodge diamond
1
00
0 2 0
0 0 0 0
0 2 30 2 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 + x1 + 27 0 x0 + 27 x1
$\mathrm{HH}^{ 3 }$ = x1 + 20 0 0 x1 + 9 11
$\mathrm{HH}^{ 4 }$ = 32 0 0 4 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 23t + 56t^{2} + 23t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(596)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(618)

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( 6t + 1 \right) \left( 2t + 1 \right) ^{4} \left( 2168t^{3} + 292t^{2} + 2t - 1 \right) \left( 7280242944t^{9} + 51172042560t^{8} + 52337034560t^{7} + 21557952224t^{6} + 4167068096t^{5} + 350307204t^{4} + 3784220t^{3} - 1237688t^{2} - 29525t + 377 \right)D^{6} + \left( 2t + 1 \right) ^{3} \left( 7954917618106368t^{15} + 58812838161874944t^{14} + 64317370487629824t^{13} + 24150065635990528t^{12} - 169207692811264t^{11} - 2851232187069952t^{10} - 939493606309632t^{9} - 141855364864128t^{8} - 10170198741184t^{7} - 161334535168t^{6} + 13625577512t^{5} - 300013508t^{4} - 20193132t^{3} + 5089982t^{2} + 126394t - 1131 \right)D^{5} + \left( 2t + 1 \right) ^{2} \left( 132581960301772800t^{16} + 1053189505210859520t^{15} + 1448429745606672384t^{14} + 830855487623915520t^{13} + 206861355700923392t^{12} + 114783163914752t^{11} - 12852970397999872t^{10} - 3250158990992256t^{9} - 350616930936640t^{8} - 9179215267808t^{7} + 1827649832272t^{6} + 196860400288t^{5} + 7480509728t^{4} + 101220344t^{3} - 5450114t^{2} - 94871t + 754 \right)D^{4} + 2 t \left( 2t + 1 \right) \left( 556844233267445760t^{16} + 4723877072314122240t^{15} + 7902601274078269440t^{14} + 5953611483813173248t^{13} + 2371187391668665344t^{12} + 474286836229374464t^{11} + 16929468530058496t^{10} - 13124169560089216t^{9} - 2915982773009344t^{8} - 236357247777440t^{7} + 1556766776448t^{6} + 1548886755776t^{5} + 75224389944t^{4} - 234350580t^{3} - 67712178t^{2} - 843381t + 1276 \right)D^{3} + 2 t \left( 2460721183200903168t^{17} + 22179978863513812992t^{16} + 43998969842694217728t^{15} + 41038532179341430784t^{14} + 21619383437263775744t^{13} + 6740267347241853952t^{12} + 1166542747324514304t^{11} + 66320421471305984t^{10} - 13874759823004800t^{9} - 3086881605643584t^{8} - 199514843776800t^{7} + 6334374121472t^{6} + 1337087620240t^{5} + 25642369276t^{4} - 1924722704t^{3} - 77890404t^{2} - 357110t + 377 \right)D^{2} + 48 t ^{2} \left( 111368846653489152t^{16} + 1006308824023035904t^{15} + 1944581595215366144t^{14} + 1764204715013677056t^{13} + 907615250888155136t^{12} + 279995837624606464t^{11} + 50172671554786944t^{10} + 4064302316406880t^{9} - 187771097529760t^{8} - 71297613176144t^{7} - 4886960565904t^{6} + 103478516062t^{5} + 18864244014t^{4} - 421424282t^{3} - 66789423t^{2} - 1849195t - 2483 \right)D + 288 t ^{3} \left( 7576112017244160t^{15} + 68570065506590720t^{14} + 130090236682129408t^{13} + 115663620687895552t^{12} + 58387479517996544t^{11} + 17775813257585920t^{10} + 3205347083920896t^{9} + 288172328884096t^{8} - 1979491304160t^{7} - 3087011373456t^{6} - 236933072704t^{5} + 318797684t^{4} + 188759612t^{3} - 52719208t^{2} - 3261448t - 71253 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ4P4 × ℙ7P7

|λ| = 20.91613 λ ≈ 20.91613; multiplicity 1; Jordan blocks 1 λ ≈ -9.458063 - 3.767907i; multiplicity 1; Jordan blocks 1 λ ≈ -9.458063 + 3.767907i; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 4; Jordan blocks 1, 3 λ ≈ 2; multiplicity 1; Jordan blocks 1 4
Sector dimension
9
Spectral radius
$20.9161262$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$20.91613$11
$-9.458063 - 3.767907\mathrm{i}$11
$-9.458063 + 3.767907\mathrm{i}$11
$-6$11
$-2$41, 3
$2$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{4} \left(\lambda + 6\right) \left(\lambda - 2\right) \left(\lambda^{3} - 2 \lambda^{2} - 292 \lambda - 2168\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda + 6\right) \left(\lambda - 2\right) \left(\lambda^{3} - 2 \lambda^{2} - 292 \lambda - 2168\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 20 & 120 & 48 & 144 & 704 & 512 & 1920 \\ 1 & -2 & 4 & 20 & 20 & 24 & 192 & 144 & 384 \\ 1 & 2 & 0 & 32 & 12 & 32 & 96 & 48 & 320 \\ 0 & 1 & 1 & -2 & 0 & 4 & 32 & 36 & 96 \\ 0 & 1 & 0 & 4 & -2 & 4 & 8 & 4 & 48 \\ 0 & 0 & 1 & 2 & 2 & -2 & 16 & 4 & 0 \\ 0 & 0 & 0 & 1 & -\frac{1}{2} & 2 & -4 & 0 & 12 \\ 0 & 0 & 0 & 1 & 2 & 0 & 8 & 2 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$