Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-432-90-d

·

A smooth divisor of bidegree $(1,1)$ in $\mathbb P^2\times\mathbb P^3$.

  • $\pi_1\colon X\to\mathbb P^3$ is a fibration whose fibers are lines, except over one point, where the fiber is a plane.

  • $\pi_2\colon X\to\mathbb P^2$ is the $\mathbb P^2$-bundle $\mathbb P_{\mathbb P^2}(\mathcal O\oplus\Omega_{\mathbb P^2}(1))$.

Varieties in this family can be blown up to

Defining data
Ambient
ℙ2P2 × ℙ3P3
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_5ω1ω2ω3ω4ω5
Bundle
E^(w1 + w3)
Bundle
O_1(1) box O_2(1)
ZL62 label
2030.a
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;4) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 90
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 432
$\chi(X, \mathrm{T}_X)$ = 12

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 432
$\mathrm{c}_1^2\mathrm{c}_2$ = 204
$\mathrm{c}_2^2$ = 97
$\mathrm{c}_1\mathrm{c}_3$ = 54
$\mathrm{c}_4$ = 9
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 3 0 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 12 0 12
$\mathrm{HH}^{ 2 }$ = 66 0 0 66
$\mathrm{HH}^{ 3 }$ = 136 0 0 0 136
$\mathrm{HH}^{ 4 }$ = 90 0 0 0 0 90
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 85t + 260t^{2} + 85t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(45)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(78)

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( 19427t^{9} + 31091t^{8} + 11664t^{7} - 1931t^{6} + 4968t^{5} - 96t^{4} + 81t^{3} + 16t^{2} - 1 \right) \left( 573247930368t^{11} - 14809363823664t^{10} + 2984039440128t^{9} + 3576837002752t^{8} + 761207952726t^{7} + 7207072776672t^{6} + 4119601560363t^{5} - 1323874349664t^{4} + 168331115520t^{3} + 38495449368t^{2} - 1904684544t + 116785152 \right)D^{6} + \left( 233866238408441856t^{20} - 6026444768198836320t^{19} - 6867628534692679968t^{18} + 1002179382555585792t^{17} + 3416144291961536530t^{16} + 4299955017661604490t^{15} + 7265035669924164507t^{14} + 3863432902879237917t^{13} - 141043738612172736t^{12} + 82569304467941625t^{11} + 332707580919055118t^{10} - 77696530063592256t^{9} + 10954259879015745t^{8} + 2148095123401986t^{7} - 43158136630080t^{6} + 69772865325075t^{5} - 10573633049088t^{4} + 1162569729024t^{3} + 238446945936t^{2} - 9523422720t + 467140608 \right)D^{5} + \left( 1948885320070348800t^{20} - 52613659179087679200t^{19} - 46415692702479905136t^{18} + 14644884091286875904t^{17} + 23659234072940814350t^{16} + 42917939317607773182t^{15} + 65667777317190304317t^{14} + 29636802965315458695t^{13} - 2532277178815528768t^{12} - 201585692335554408t^{11} + 1832838780075942672t^{10} - 338885922914371728t^{9} + 43947393062386934t^{8} + 14760056853150246t^{7} - 1000951734768288t^{6} + 7101264751119t^{5} + 16996257311232t^{4} - 1808382081024t^{3} - 338008046040t^{2} + 9432723456t - 467140608 \right)D^{4} + t ^{2} \left( 8185318344295464960t^{18} - 228875212717689595200t^{17} - 156284293672200299040t^{16} + 76847064253974273792t^{15} + 79273043445552437686t^{14} + 200490566115197018142t^{13} + 300054663942687201869t^{12} + 114463683596811337035t^{11} - 15528695643566400064t^{10} - 1042358503677647517t^{9} + 5632832358895315096t^{8} - 650102095993099680t^{7} + 92998626122557416t^{6} + 46784450573619174t^{5} - 2767822665203904t^{4} + 241445389149369t^{3} - 8677096759392t^{2} + 768448954368t + 12457082880 \right)D^{3} + t ^{2} \left( 18085655770252836864t^{18} - 519056349427466898960t^{17} - 276270042187459479744t^{16} + 188641820662566166784t^{15} + 130858754775397997936t^{14} + 483002651372463255120t^{13} + 717855559869085732274t^{12} + 230522686888976957634t^{11} - 42475808724074251200t^{10} + 560482208184375627t^{9} + 10224214229895228992t^{8} - 492757644179670336t^{7} + 97518635020005440t^{6} + 76097350461935592t^{5} - 4853767442327424t^{4} + 374979104453520t^{3} - 4625104293504t^{2} + 941749493760t + 3737124864 \right)D^{2} + 2 t ^{3} \left( 9822382013154557952t^{17} - 287215007342738741040t^{16} - 121546578335569212096t^{15} + 107936105930700126720t^{14} + 49382249367719441684t^{13} + 282111281904640266756t^{12} + 416502857929843870196t^{11} + 113658391896357143304t^{10} - 26126330063614464240t^{9} + 2273503681553343615t^{8} + 4981502236638997656t^{7} + 3259480127762232t^{6} + 21157527293533584t^{5} + 30905765920818072t^{4} - 2025470276585856t^{3} + 163248295919616t^{2} + 853957225152t + 271103459328 \right)D + 24 t ^{3} \left( 334094626297774080t^{17} - 9891324371050503600t^{16} - 3465148473146402496t^{15} + 3768597088449347712t^{14} + 1114310482715527436t^{13} + 10123603568225565948t^{12} + 14853318985415666120t^{11} + 3553462904939037396t^{10} - 954685308991985928t^{9} + 133752146080603347t^{8} + 160322264177359176t^{7} + 4867952794110648t^{6} + 137389289683248t^{5} + 826573293143406t^{4} - 54242610234048t^{3} + 4789040578560t^{2} + 70915360032t + 5025411072 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ3P3

|λ| = 6.605047 λ ≈ 6.605047; multiplicity 1; Jordan blocks 1 λ ≈ -4.907822 - 3.447352i; multiplicity 1; Jordan blocks 1 λ ≈ -4.907822 + 3.447352i; multiplicity 1; Jordan blocks 1 λ ≈ 1.331516 - 4.164144i; multiplicity 1; Jordan blocks 1 λ ≈ 1.331516 + 4.164144i; multiplicity 1; Jordan blocks 1 λ ≈ 1.08681 - 2.036968i; multiplicity 1; Jordan blocks 1 λ ≈ 1.08681 + 2.036968i; multiplicity 1; Jordan blocks 1 λ ≈ -0.8130279 - 0.3762535i; multiplicity 1; Jordan blocks 1 λ ≈ -0.8130279 + 0.3762535i; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$6.60504694$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$6.605047$11
$-4.907822 - 3.447352\mathrm{i}$11
$-4.907822 + 3.447352\mathrm{i}$11
$1.331516 - 4.164144\mathrm{i}$11
$1.331516 + 4.164144\mathrm{i}$11
$1.08681 - 2.036968\mathrm{i}$11
$1.08681 + 2.036968\mathrm{i}$11
$-0.8130279 - 0.3762535\mathrm{i}$11
$-0.8130279 + 0.3762535\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\lambda^{9} - 16 \lambda^{7} - 81 \lambda^{6} + 96 \lambda^{5} - 4968 \lambda^{4} + 1931 \lambda^{3} - 11664 \lambda^{2} - 31091 \lambda - 19427\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{9} - 16 \lambda^{7} - 81 \lambda^{6} + 96 \lambda^{5} - 4968 \lambda^{4} + 1931 \lambda^{3} - 11664 \lambda^{2} - 31091 \lambda - 19427\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 0 & 0 & 0 & 3 & 0 & 0 & 5 \\ 2 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 \\ 3 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 \\ 0 & 3 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 3 & 0 & 0 & 0 & 2 & 2 & 0 \\ 0 & 0 & 0 & 3 & 0 & 5 & 0 & 0 & 2 \\ 0 & 0 & 0 & 2 & 3 & -3 & 0 & 0 & -2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 5 & 3 & 0\end{pmatrix}\notag\end{equation}$$