Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-44-16-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ3P3
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
E^(w1 + w2 + 3w3)
Bundle
O_1(1) box O_2(1) box O_3(3)
ZL62 label
1130.x4002
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;4) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 16
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 44
$\chi(X, \mathrm{T}_X)$ = -58

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 44
$\mathrm{c}_1^2\mathrm{c}_2$ = 92
$\mathrm{c}_2^2$ = 180
$\mathrm{c}_1\mathrm{c}_3$ = -28
$\mathrm{c}_4$ = 116
Hodge diamond
1
00
0 3 0
0 0 0 0
0 10 88 10 0
0 0 0 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = 58 0 58 0
$\mathrm{HH}^{ 3 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 4 }$ = 57 0 0 41 0 16
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 11t + 20t^{2} + 11t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(728)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(721)

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( 9t + 1 \right) \left( 19t - 1 \right) \left( 8t + 1 \right) ^{4} \left( 472t^{2} + 55t - 1 \right) \left( 4520688384t^{6} + 2152228128t^{5} + 344393728t^{4} + 19630026t^{3} + 29810t^{2} - 17741t + 11 \right)D^{6} + \left( 8t + 1 \right) ^{3} \left( 61298798542700544t^{11} + 42176304845875200t^{10} + 11267316547680000t^{9} + 1430007783915296t^{8} + 77581626424848t^{7} - 38146622786t^{6} - 138512214846t^{5} - 1723412439t^{4} + 42798489t^{3} - 477293t^{2} + 52761t - 22 \right)D^{5} + \left( 8t + 1 \right) ^{2} \left( 4086586569513369600t^{12} + 3034735619326279680t^{11} + 904592953864200192t^{10} + 136931715759913984t^{9} + 10667849835049952t^{8} + 333040836386256t^{7} - 4967054395182t^{6} - 454787623388t^{5} + 941349321t^{4} + 334380108t^{3} + 1008498t^{2} - 33192t + 11 \right)D^{4} + t \left( 8t + 1 \right) \left( 137309308735649218560t^{12} + 111531768543398461440t^{11} + 37377858515705856000t^{10} + 6649853086665064448t^{9} + 663613715966163712t^{8} + 34525316969648096t^{7} + 538760169150064t^{6} - 27380954893118t^{5} - 1105363702154t^{4} - 2739721121t^{3} + 149813077t^{2} - 1405179t - 145 \right)D^{3} + 2 t ^{2} \left( 1213552747682690236416t^{12} + 1086038691452241838080t^{11} + 410238697367654105088t^{10} + 84972115850654154752t^{9} + 10408051505298735104t^{8} + 743347677006827264t^{7} + 26297280665556736t^{6} + 16284207226560t^{5} - 30291333862698t^{4} - 766220471662t^{3} - 1957589702t^{2} + 22517059t - 341297 \right)D^{2} + 16 t ^{2} \left( 164771170482779062272t^{12} + 142345181515733532672t^{11} + 51531722571711873024t^{10} + 10144256262176800768t^{9} + 1167151956672101888t^{8} + 76604558131271744t^{7} + 2313037933330840t^{6} - 18265047628944t^{5} - 3236298658371t^{4} - 68274415306t^{3} - 169551314t^{2} - 533540t - 10021 \right)D + 128 t ^{2} \left( 8406692371570360320t^{12} + 7086279857993809920t^{11} + 2488461620015775744t^{10} + 471813274388908032t^{9} + 51741521560267776t^{8} + 3170445201565120t^{7} + 82396222793032t^{6} - 1387600776184t^{5} - 136707347145t^{4} - 2507646656t^{3} - 6490916t^{2} - 66247t - 176 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ3P3

|λ| = 62.5464 λ ≈ 62.5464; multiplicity 1; Jordan blocks 1 λ ≈ 19; multiplicity 2; Jordan blocks 1, 1 λ ≈ -9; multiplicity 1; Jordan blocks 1 λ ≈ -8; multiplicity 7; Jordan blocks 1, 1, 2, 3 λ ≈ -7.546398; multiplicity 1; Jordan blocks 1 2 7
Sector dimension
12
Spectral radius
$62.5463978$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$62.5464$11
$19$21, 1
$-9$11
$-8$71, 1, 2, 3
$-7.546398$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 8\right)^{7} \left(\lambda + 9\right) \left(\lambda - 19\right)^{2} \left(\lambda^{2} - 55 \lambda - 472\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 8\right)^{3} \left(\lambda + 9\right) \left(\lambda - 19\right) \left(\lambda^{2} - 55 \lambda - 472\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 38 & 38 & 180 & 216 & 1242 & 1242 & 2688 & 9792 & 19080 & 19080 & 144120 \\ 1 & -1 & 1 & 11 & 36 & 174 & 66 & 216 & 1206 & 2310 & 1554 & 15816 \\ 1 & 1 & -1 & 11 & 36 & 66 & 174 & 216 & 1206 & 1554 & 2310 & 15816 \\ 1 & 3 & 3 & 9 & 6 & 92 & 92 & 228 & 540 & 1386 & 1386 & 9792 \\ 0 & 1 & 1 & 0 & -2 & 11 & 11 & 12 & 168 & 200 & 200 & 1908 \\ 0 & 1 & 0 & 1 & 3 & 8 & 1 & 14 & 90 & 192 & 84 & 1170 \\ 0 & 0 & 1 & 1 & 3 & 1 & 8 & 14 & 90 & 84 & 192 & 1170 \\ 0 & 0 & 0 & 1 & 0 & 3 & 3 & 0 & 6 & 38 & 38 & 216 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & -\frac{2}{9} & 7 & \frac{40}{3} & \frac{40}{3} & \frac{464}{3} \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{4}{3} & 3 & 0 & 2 & 38 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{4}{3} & 3 & 2 & 0 & 38 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{4}{3} & \frac{4}{3} & 0\end{pmatrix}\notag\end{equation}$$