Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-80-23-a

·

This is a Fano fourfold of K3 type, appearing as case K3-37 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ5P5
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_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
2 copies of E^(2w3) + E^(w1 + w2 + w3)
Bundle
2 copies of O_3(2) + O_1(1) box O_2(1) box O_3(1)
ZL62 label
1150.99x4000
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;6) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 23
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 80
$\chi(X, \mathrm{T}_X)$ = -20

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 80
$\mathrm{c}_1^2\mathrm{c}_2$ = 104
$\mathrm{c}_2^2$ = 128
$\mathrm{c}_1\mathrm{c}_3$ = 24
$\mathrm{c}_4$ = 24
Hodge diamond
1
00
0 3 0
0 2 2 0
0 1 22 1 0
0 2 2 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 + x2 + 20 0 x0 + 20 x2
$\mathrm{HH}^{ 3 }$ = x1 + 2 0 0 x1 2
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 5 0 0 x1 - x2 - 20 2 23
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 18t + 42t^{2} + 18t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(673)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(673)

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( 10t - 1 \right) \left( 30t - 1 \right) \left( 2t + 1 \right) ^{3} \left( 6t + 1 \right) ^{4} \left( 881156160t^{8} + 1165756800t^{7} + 575727328t^{6} + 129561696t^{5} + 13333396t^{4} + 416112t^{3} - 28896t^{2} - 2884t + 5 \right)D^{6} + 2 \left( 2t + 1 \right) ^{2} \left( 6t + 1 \right) ^{3} \left( 33307702848000t^{12} + 53407072915200t^{11} + 33459985975680t^{10} + 10119391680960t^{9} + 1457317769760t^{8} + 70995309712t^{7} - 4301723544t^{6} - 652925980t^{5} - 26432244t^{4} + 1690652t^{3} + 2494t^{2} + 4331t - 5 \right)D^{5} + \left( 2t + 1 \right) \left( 6t + 1 \right) ^{2} \left( 6661540569600000t^{14} + 13774818859776000t^{13} + 11867983520762880t^{12} + 5489879948605440t^{11} + 1467020323226112t^{10} + 225891231081472t^{9} + 18034644858240t^{8} + 307954016576t^{7} - 67040166832t^{6} - 4616115256t^{5} + 119111500t^{4} + 12490228t^{3} + 35692t^{2} - 5352t + 5 \right)D^{4} + 2 t \left( 6t + 1 \right) \left( 167870822353920000t^{15} + 432356385996288000t^{14} + 481316374838016000t^{13} + 302865560122936320t^{12} + 118379354299683840t^{11} + 29751500711290368t^{10} + 4790516869539584t^{9} + 465430246121600t^{8} + 20382282701024t^{7} - 801690396688t^{6} - 160083561480t^{5} - 7264587012t^{4} - 10280496t^{3} + 2907448t^{2} - 57782t - 13 \right)D^{3} + 4 t ^{2} \left( 1112743736745984000t^{15} + 2916041898804940800t^{14} + 3340564322843105280t^{13} + 2196220602622602240t^{12} + 915828972070869504t^{11} + 252816967492821504t^{10} + 46661099695372416t^{9} + 5602805543849280t^{8} + 386361379001664t^{7} + 5226709542224t^{6} - 1676761027872t^{5} - 158387832744t^{4} - 5485391506t^{3} - 34804350t^{2} + 358701t - 16090 \right)D^{2} + 16 t ^{2} \left( 302167480237056000t^{15} + 764927565632870400t^{14} + 844076685253800960t^{13} + 532654528851302400t^{12} + 212354656391450880t^{11} + 55786730387120640t^{10} + 9734740192612896t^{9} + 1089924083846048t^{8} + 66405392273904t^{7} - 124967037240t^{6} - 393669443922t^{5} - 31998749594t^{4} - 1037840363t^{3} - 9456980t^{2} - 49564t - 1195 \right)D + 32 t ^{2} \left( 61666832701440000t^{15} + 152394828212736000t^{14} + 163744140562053120t^{13} + 100293792113134080t^{12} + 38665160567059968t^{11} + 9778695836875776t^{10} + 1631964251539296t^{9} + 172233093622656t^{8} + 9291509996640t^{7} - 174966992096t^{6} - 71297417466t^{5} - 5216552190t^{4} - 160010693t^{3} - 1688178t^{2} - 16647t - 95 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × Gr(2, 4)

|λ| = 30 λ ≈ 30; multiplicity 1; Jordan blocks 1 λ ≈ 10; multiplicity 2; Jordan blocks 1, 1 λ ≈ -6; multiplicity 6; Jordan blocks 1, 2, 3 λ ≈ -2; multiplicity 3; Jordan blocks 1, 2 2 6 3
Sector dimension
12
Spectral radius
$30$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$30$11
$10$21, 1
$-6$61, 2, 3
$-2$31, 2

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda + 6\right)^{6} \left(\lambda - 10\right)^{2} \left(\lambda - 30\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 6\right)^{3} \left(\lambda - 10\right) \left(\lambda - 30\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 10 & 10 & 56 & 24 & 144 & 144 & 240 & 480 & 688 & 688 & 2080 \\ 1 & -1 & 1 & 4 & 8 & 48 & 8 & 28 & 120 & 180 & 84 & 448 \\ 1 & 1 & -1 & 4 & 8 & 8 & 48 & 28 & 120 & 84 & 180 & 448 \\ 1 & 1 & 1 & 4 & 2 & 18 & 18 & 36 & 48 & 84 & 84 & 240 \\ 0 & 1 & 1 & 0 & -2 & 4 & 4 & 0 & 40 & 28 & 28 & 120 \\ 0 & 1 & 0 & 1 & 1 & 3 & 1 & 4 & 16 & 28 & 8 & 60 \\ 0 & 0 & 1 & 1 & 1 & 1 & 3 & 4 & 16 & 8 & 28 & 60 \\ 0 & 0 & 0 & 2 & 0 & 2 & 2 & 0 & 4 & 10 & 10 & 24 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & -1 & 2 & 3 & 3 & 18 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 2 & 0 & 2 & 10 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 2 & 0 & 10 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$