Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 4-228-51-a

·

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

Defining data
Ambient
Fl(1, 3, 4) × Gr(2, 4)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w5) + E^(w2 + w4)
Bundle
O_2(1) + U_1,2^v ot U_1,3^v box U_2^v
ZL62 label
3134.1j
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,3;4) \\ \det\mathcal{E} &\cong \mathcal{O}(2,2;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 51
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 228
$\chi(X, \mathrm{T}_X)$ = 2

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 228
$\mathrm{c}_1^2\mathrm{c}_2$ = 144
$\mathrm{c}_2^2$ = 112
$\mathrm{c}_1\mathrm{c}_3$ = 56
$\mathrm{c}_4$ = 20
Hodge diamond
1
00
0 4 0
0 0 0 0
0 0 10 0 0
0 0 0 0
0 4 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 - 2 0 x0 - 2 x1
$\mathrm{HH}^{ 3 }$ = x1 + 30 0 0 x1 - 16 46
$\mathrm{HH}^{ 4 }$ = 51 0 0 0 0 51
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 46t + 134t^{2} + 46t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(280)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(359)

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( 6t + 1 \right) \left( 7t + 1 \right) \left( 10t - 1 \right) \left( t - 1 \right) ^{2} \left( 2t - 1 \right) ^{2} \left( 20t^{2} + 4t + 1 \right) \left( 363734428160t^{10} - 238698036480t^{9} - 229694754816t^{8} + 213642873472t^{7} - 49222066752t^{6} + 14297729152t^{5} - 2378886288t^{4} - 277532040t^{3} - 18147134t^{2} - 55523t - 4161 \right)D^{6} + \left( t - 1 \right) \left( 2t - 1 \right) \left( 128325506254848000t^{17} - 180116660213350400t^{16} - 32939913841684480t^{15} + 158713945014190080t^{14} - 81217200350397440t^{13} + 15896726655541248t^{12} - 3750673068224256t^{11} + 291974570047488t^{10} + 175665204935104t^{9} + 25379370822016t^{8} - 6223316206480t^{7} - 1671703986432t^{6} - 81347759580t^{5} - 24179811136t^{4} - 1576110357t^{3} - 74874484t^{2} - 166569t - 8322 \right)D^{5} + \left( 2138758437580800000t^{19} - 5622989776480256000t^{18} + 3737986541987880960t^{17} + 2615036048410685440t^{16} - 5180743921834805248t^{15} + 3156198919164892160t^{14} - 1048735219396353536t^{13} + 248207406984670464t^{12} - 37428385873297792t^{11} - 1911341095025600t^{10} + 874079916023072t^{9} + 165451110821328t^{8} - 19922290659352t^{7} - 1218964126516t^{6} - 364118081666t^{5} + 11040429571t^{4} + 1183713200t^{3} + 70240192t^{2} + 249162t + 4161 \right)D^{4} + t \left( 8982785437839360000t^{18} - 21679152745842688000t^{17} + 11777899542029312000t^{16} + 12633386664685383680t^{15} - 20253441202446878720t^{14} + 11782576448377230336t^{13} - 4060666991980398080t^{12} + 1038732205521906432t^{11} - 157953659769917568t^{10} - 5003622007199168t^{9} + 2968641767977056t^{8} + 341302430244624t^{7} - 17856658274648t^{6} - 3021645546068t^{5} - 204673907642t^{4} - 8368948935t^{3} - 210141541t^{2} - 6370635t - 803 \right)D^{3} + 2 t ^{2} \left( 9923839150374912000t^{17} - 22312734147888332800t^{16} + 9895378168271298560t^{15} + 15057524710344253440t^{14} - 21024782242952738816t^{13} + 11807051961939077120t^{12} - 4167556278699142656t^{11} + 1104638847882993408t^{10} - 165943030890036736t^{9} - 2975552750870976t^{8} + 2335732589140000t^{7} + 256342026608896t^{6} + 1441255492776t^{5} - 1465066629632t^{4} - 56898069754t^{3} - 2783610621t^{2} - 122824271t - 2548698 \right)D^{2} + 8 t ^{2} \left( 2694835631351808000t^{17} - 5733187289228902400t^{16} + 2094307533055672320t^{15} + 4267959496008591360t^{14} - 5430303384237045760t^{13} + 2970160339310044160t^{12} - 1064414530472347392t^{11} + 285802904254037760t^{10} - 41754168312113664t^{9} - 489548473045440t^{8} + 444994937797936t^{7} + 57537780610072t^{6} + 2458520314404t^{5} - 220101268408t^{4} - 3943581312t^{3} - 339737499t^{2} - 18174231t - 274626 \right)D + 96 t ^{2} \left( 91661075896320000t^{17} - 187517949249536000t^{16} + 57995971831357440t^{15} + 148728349662781440t^{14} - 178151806432361472t^{13} + 95664012368999936t^{12} - 34601841401164032t^{11} + 9317445762758912t^{10} - 1321079811399040t^{9} - 12781317755712t^{8} + 11078890602064t^{7} + 1767943690960t^{6} + 106753208472t^{5} - 4849060206t^{4} - 37455134t^{3} - 7929516t^{2} - 372811t - 4161 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 4) × Gr(2, 4)

|λ| = 10 λ ≈ 10; multiplicity 1; Jordan blocks 1 λ ≈ -7; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 1; Jordan blocks 1 λ ≈ -2 - 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2 + 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 2; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ 1; multiplicity 3; Jordan blocks 1, 1, 1 2 2 3 3
Sector dimension
13
Spectral radius
$10$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$10$11
$-7$11
$-6$11
$-2 - 4\mathrm{i}$21, 1
$-2 + 4\mathrm{i}$21, 1
$2$31, 1, 1
$1$31, 1, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 6\right) \left(\lambda + 7\right) \left(\lambda - 1\right)^{3} \left(\lambda - 10\right) \left(\lambda - 2\right)^{3} \left(\lambda^{2} + 4 \lambda + 20\right)^{2}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 6\right) \left(\lambda + 7\right) \left(\lambda - 1\right) \left(\lambda - 10\right) \left(\lambda - 2\right) \left(\lambda^{2} + 4 \lambda + 20\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 4 & 6 & 6 & 6 & 6 & 6 & 4 & 24 & 16 & 0 \\ 1 & 0 & 0 & 1 & 2 & 0 & 2 & 2 & 4 & -3 & 0 & 0 & 4 \\ 1 & 0 & 0 & 1 & 2 & 4 & 2 & 2 & 0 & 3 & 0 & 0 & 4 \\ 1 & 0 & 0 & -1 & 2 & 4 & 2 & 2 & 4 & 3 & 12 & 6 & 0 \\ 0 & 1 & 1 & -4 & 0 & 1 & -4 & -4 & 1 & -2 & -12 & -8 & -6 \\ 0 & 1 & 0 & 3 & 0 & -1 & 0 & 2 & 0 & 0 & 6 & 4 & 3 \\ 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 2 & 4 & 2 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 4 & 2 & 0 \\ 0 & 0 & 1 & 3 & 0 & 0 & 2 & 0 & -1 & 2 & 6 & 4 & 3 \\ 0 & 0 & 0 & 0 & 0 & -2 & -4 & 4 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 4 & 4 & 6 & 2 & 2 & 0 & 2 & 2 & 0 \\ 0 & 0 & 0 & 0 & -3 & -1 & -6 & -2 & 1 & 0 & -3 & -3 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 10 & 6 & 0\end{pmatrix}\notag\end{equation}$$