Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-100-27-d

·

This is a Fano fourfold of K3 type, appearing as case K3-47 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ1P1 × Fl(1, 3, 4)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)ω1ω2ω3ω4
Bundle
E^(w2 + w4) + E^(w1 + w2 + w4)
Bundle
O_2(1,1) + O_1(1) box O_2(1,1)
ZL62 label
134.Wx4100
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;3,3) \\ \det\mathcal{E} &\cong \mathcal{O}(1;2,2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 27
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 100
$\chi(X, \mathrm{T}_X)$ = -22

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 100
$\mathrm{c}_1^2\mathrm{c}_2$ = 112
$\mathrm{c}_2^2$ = 128
$\mathrm{c}_1\mathrm{c}_3$ = 24
$\mathrm{c}_4$ = 36
Hodge diamond
1
00
0 3 0
0 0 0 0
0 1 26 1 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 + x2 + 22 0 x0 + 22 x2
$\mathrm{HH}^{ 3 }$ = x1 + 6 0 0 x1 6
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 19 0 0 x1 - x2 - 8 0 27
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 22t + 54t^{2} + 22t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(601)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(621)

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( 22t - 1 \right) \left( 2t + 1 \right) ^{2} \left( 3t + 1 \right) ^{3} \left( 52t^{2} + 12t + 1 \right) \left( 2122854912t^{9} + 3563640576t^{8} + 2430433536t^{7} + 858474880t^{6} + 142910592t^{5} + 2556320t^{4} - 1644240t^{3} - 50824t^{2} + 822t - 5 \right)D^{6} + \left( 2t + 1 \right) \left( 3t + 1 \right) ^{2} \left( 1835980790611968t^{15} + 4218249633988608t^{14} + 4148354221830144t^{13} + 2266440785731584t^{12} + 720102940769280t^{11} + 116579394897920t^{10} + 1868214807808t^{9} - 2198665364992t^{8} - 207658078144t^{7} + 10669427712t^{6} + 1257890448t^{5} - 16450656t^{4} + 6044500t^{3} + 219736t^{2} - 2461t + 10 \right)D^{5} + \left( 3t + 1 \right) \left( 91799039530598400t^{17} + 270290233095045120t^{16} + 352294860609773568t^{15} + 267161138767024128t^{14} + 128855582257416192t^{13} + 39920703785628672t^{12} + 7384195346427392t^{11} + 575860668235008t^{10} - 46444897366912t^{9} - 11959312694080t^{8} - 323605572832t^{7} + 71140237552t^{6} + 5107235064t^{5} + 61313244t^{4} - 11042998t^{3} - 210337t^{2} + 1186t - 5 \right)D^{4} + t \left( 1156667898085539840t^{17} + 3621710565715230720t^{16} + 5083068516300103680t^{15} + 4215180956941651968t^{14} + 2271833525785208832t^{13} + 815952109091484672t^{12} + 189379464638043648t^{11} + 24771924356108032t^{10} + 696804493297024t^{9} - 283391131945024t^{8} - 37635774615200t^{7} - 1075590292240t^{6} + 62639925848t^{5} + 2483768956t^{4} - 19078126t^{3} - 254263t^{2} + 42078t + 13 \right)D^{3} + 2 t ^{2} \left( 1277842630265929728t^{16} + 3857858030348845056t^{15} + 5214554576480231424t^{14} + 4158949624363892736t^{13} + 2147386413584037888t^{12} + 732013785736833024t^{11} + 157947364940456448t^{10} + 18010767235229440t^{9} + 6042553856256t^{8} - 276883232281920t^{7} - 31035104684384t^{6} - 1018549454048t^{5} + 13715748664t^{4} + 800574848t^{3} - 7589722t^{2} + 707915t + 13850 \right)D^{2} + 8 t ^{2} \left( 347000369425661952t^{16} + 1019097467375026176t^{15} + 1338154683415805952t^{14} + 1035110661011527680t^{13} + 516147949758062592t^{12} + 168275821613184000t^{11} + 33962931945597696t^{10} + 3343848588816128t^{9} - 109623588487936t^{8} - 67142645870784t^{7} - 6588490155696t^{6} - 218539813112t^{5} + 252178388t^{4} + 112888448t^{3} - 651756t^{2} + 148339t + 1190 \right)D + 288 t ^{2} \left( 3934244551311360t^{16} + 11336007235092480t^{15} + 14584421652406272t^{14} + 11036237679470592t^{13} + 5362739338567680t^{12} + 1689093068702208t^{11} + 322647530386944t^{10} + 27567237593856t^{9} - 1889477083200t^{8} - 692077073280t^{7} - 61539589952t^{6} - 1988522552t^{5} - 4854216t^{4} + 935602t^{3} + 3240t^{2} + 1114t + 5 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × Fl(1, 3, 4)

|λ| = 22 λ ≈ 22; multiplicity 1; Jordan blocks 1 λ ≈ -6 - 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -6 + 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 6; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 4; Jordan blocks 1, 1, 2 λ ≈ -2; multiplicity 3; Jordan blocks 1, 1, 1 2 2 4 3
Sector dimension
13
Spectral radius
$22$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$22$11
$-6 - 4\mathrm{i}$21, 1
$-6 + 4\mathrm{i}$21, 1
$6$11
$-3$41, 1, 2
$-2$31, 1, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda + 3\right)^{4} \left(\lambda - 22\right) \left(\lambda - 6\right) \left(\lambda^{2} + 12 \lambda + 52\right)^{2}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right)^{2} \left(\lambda - 22\right) \left(\lambda - 6\right) \left(\lambda^{2} + 12 \lambda + 52\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 16 & 16 & 42 & 42 & 114 & 78 & 78 & 312 & 208 & 132 & 320 \\ 1 & -1 & 1 & 1 & 14 & 14 & 20 & 14 & 14 & 90 & 60 & 27 & 80 \\ 1 & 1 & -1 & 3 & 2 & 6 & 20 & 14 & 14 & 54 & 42 & 21 & 52 \\ 1 & 1 & 3 & -1 & 6 & 2 & 20 & 14 & 14 & 54 & 30 & 27 & 52 \\ 0 & 1 & 1 & 0 & -2 & 3 & 1 & 0 & 2 & 18 & 14 & 4 & 15 \\ 0 & 1 & 0 & 1 & 3 & -2 & 1 & 2 & 0 & 18 & 10 & 6 & 15 \\ 0 & 0 & 1 & 1 & 1 & 1 & -2 & 0 & 0 & 0 & 0 & 2 & 6 \\ 0 & 0 & 1 & 0 & 1 & 0 & 3 & -1 & 3 & 6 & 2 & 4 & 6 \\ 0 & 0 & 0 & 1 & 0 & 1 & 3 & 3 & -1 & 6 & 6 & 2 & 6 \\ 0 & 0 & 0 & 0 & 1 & \frac{7}{3} & -1 & -2 & \frac{2}{3} & -1 & 2 & -\frac{2}{3} & \frac{10}{3} \\ 0 & 0 & 0 & 0 & 1 & -1 & 3 & 4 & 0 & 3 & -2 & 2 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 6 & 4 & 4 & 6 & 4 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 4 & 2 & 0\end{pmatrix}\notag\end{equation}$$