Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-260-58-b

·

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

Defining data
Ambient
ℙ4P4 × 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)node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9 (marked)ω1ω2ω3ω4ω5ω6ω7ω8ω9
Bundle
E^(w4 + w5) + E^(w4 + w9) + E^(2w1)
Bundle
Q_1 box O_2(1,0) + Q_1 box O_2(0,1) + O_1(2)
ZL62 label
405G.wPwL9
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;5,5) \\ \det\mathcal{E} &\cong \mathcal{O}(4;4,4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 58
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 260
$\chi(X, \mathrm{T}_X)$ = -2

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 260
$\mathrm{c}_1^2\mathrm{c}_2$ = 164
$\mathrm{c}_2^2$ = 96
$\mathrm{c}_1\mathrm{c}_3$ = 52
$\mathrm{c}_4$ = 16
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 8 0 0
0 0 0 0
0 3 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 + 45 0 0 x1 - 15 60
$\mathrm{HH}^{ 4 }$ = 58 0 0 0 0 58
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 53t + 152t^{2} + 53t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(119)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(164)

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( 2t + 1 \right) ^{2} \left( 20t^{2} + 4t + 1 \right) \left( 440t^{3} - 12t^{2} + 6t - 1 \right) \left( 134902699520t^{9} - 100553744384t^{8} - 91426781568t^{7} - 13293479104t^{6} + 569623904t^{5} + 120122448t^{4} - 2119496t^{3} - 720452t^{2} - 13428t - 65 \right)D^{6} + 2 \left( 2t + 1 \right) \left( 49860037742592000t^{16} - 36711213337395200t^{15} - 43338900847165440t^{14} - 4891897701965824t^{13} + 4139964738926592t^{12} + 1368810037017600t^{11} + 155495419006976t^{10} + 1134202720256t^{9} - 1906131186048t^{8} + 183570208704t^{7} + 68381764480t^{6} + 333996000t^{5} - 334547608t^{4} + 2658180t^{3} + 1400880t^{2} + 20012t + 65 \right)D^{5} + \left( 1662001258086400000t^{17} - 610272822214656000t^{16} - 1997938102246031360t^{15} - 801814352052396032t^{14} + 4261040562937856t^{13} + 59512648030918656t^{12} + 11045655475303424t^{11} + 200294248323072t^{10} - 108762657201152t^{9} - 11776761211904t^{8} - 1377985462080t^{7} - 76662959264t^{6} + 14796172960t^{5} + 823483160t^{4} - 16044420t^{3} - 2613892t^{2} - 30534t - 65 \right)D^{4} + 2 t \left( 3490202641981440000t^{16} - 1640557387669504000t^{15} - 4104437503756697600t^{14} - 1556184438800097280t^{13} - 50765768565342208t^{12} + 67660764123573248t^{11} + 9772584320648704t^{10} - 438102781262336t^{9} - 143621921083392t^{8} + 1960564820352t^{7} + 2118290232512t^{6} + 206542901728t^{5} + 11264981600t^{4} + 262748684t^{3} + 2585846t^{2} + 10846t + 19 \right)D^{3} + 4 t ^{2} \left( 3855842918760448000t^{15} - 2115852646524518400t^{14} - 4460640871736913920t^{13} - 1607603468133320704t^{12} - 85730070227294208t^{11} + 44409875106159616t^{10} + 5118294077513728t^{9} - 446723618532864t^{8} - 61813470963456t^{7} + 9278714581120t^{6} + 1982327113296t^{5} + 138389712184t^{4} + 5738566760t^{3} + 124765164t^{2} + 1486894t + 6955 \right)D^{2} + 16 t ^{2} \left( 1047060792594432000t^{15} - 634945126933299200t^{14} - 1197070617439037440t^{13} - 412921859767593984t^{12} - 26117412837042176t^{11} + 8131068612748288t^{10} + 780656343898496t^{9} - 79776088398912t^{8} - 2444089206432t^{7} + 2951415365904t^{6} + 422338660288t^{5} + 24373774832t^{4} + 827242770t^{3} + 16958815t^{2} + 191431t + 845 \right)D + 32 t ^{2} \left( 213685876039680000t^{15} - 137905880752128000t^{14} - 242377535594065920t^{13} - 80852714952689664t^{12} - 5459681881189376t^{11} + 1274872319324672t^{10} + 108684757606400t^{9} - 9878263762944t^{8} + 954172625280t^{7} + 604010608704t^{6} + 69270065808t^{5} + 3514943736t^{4} + 103378692t^{3} + 2043994t^{2} + 19454t + 65 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ4P4 × Fl(1, 5, 6)

|λ| = 9.559526 λ ≈ 9.559526; multiplicity 1; Jordan blocks 1 λ ≈ -1.779763 - 6.546742i; multiplicity 1; Jordan blocks 1 λ ≈ -1.779763 + 6.546742i; 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, 2 λ ≈ 2; multiplicity 1; Jordan blocks 1 2 2 3
Sector dimension
11
Spectral radius
$9.5595263$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$9.559526$11
$-1.779763 - 6.546742\mathrm{i}$11
$-1.779763 + 6.546742\mathrm{i}$11
$-2 - 4\mathrm{i}$21, 1
$-2 + 4\mathrm{i}$21, 1
$-2$31, 2
$2$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda - 2\right) \left(\lambda^{2} + 4 \lambda + 20\right)^{2} \left(\lambda^{3} - 6 \lambda^{2} + 12 \lambda - 440\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda - 2\right) \left(\lambda^{2} + 4 \lambda + 20\right) \left(\lambda^{3} - 6 \lambda^{2} + 12 \lambda - 440\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 2 & 2 & 12 & 12 & 12 & 32 & 16 & 16 & 0 \\ 1 & 0 & 1 & 1 & -2 & -2 & -8 & 0 & 6 & 6 & 32 \\ 1 & 0 & -1 & 1 & 0 & 4 & 4 & 12 & 6 & 6 & 0 \\ 1 & 0 & 1 & -1 & 4 & 0 & 4 & 12 & 6 & 6 & 0 \\ 0 & 1 & 3 & 1 & -1 & 1 & 0 & 0 & 0 & 4 & 12 \\ 0 & 1 & 1 & 3 & 1 & -1 & 0 & 0 & 4 & 0 & 12 \\ 0 & 1 & -1 & -1 & 1 & 1 & 0 & 4 & -2 & -2 & -12 \\ 0 & 0 & 0 & 0 & 1 & 1 & -1 & -2 & -1 & -1 & -4 \\ 0 & 0 & 0 & 0 & 2 & 0 & 2 & 2 & 0 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 2 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$