Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-211-49-a

·

This is a Fano fourfold of K3 type, appearing as case C-6 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
Fl(1, 2, 6)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
E^(w5) + E^(w1 + 2w2)
Bundle
wedge^3 Q_1^v ot U_2^v ot U_3^v + O(1,2)
ZL62 label
52.4x301
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,5) \\ \det\mathcal{E} &\cong \mathcal{O}(1,3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 49
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 211
$\chi(X, \mathrm{T}_X)$ = -19

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 211
$\mathrm{c}_1^2\mathrm{c}_2$ = 154
$\mathrm{c}_2^2$ = 106
$\mathrm{c}_1\mathrm{c}_3$ = 26
$\mathrm{c}_4$ = 29
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 21 1 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 + 19 0 x0 + 19 x1
$\mathrm{HH}^{ 3 }$ = x1 + 41 0 0 x1 - 10 51
$\mathrm{HH}^{ 4 }$ = 50 0 0 1 0 49
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 44t + 121t^{2} + 44t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(34)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(39)

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( t + 1 \right) ^{3} \left( 14789t^{5} + 3750t^{4} + 825t^{3} - 16t^{2} - 1 \right) \left( 26923329888t^{6} + 25801943712t^{5} + 15295153814t^{4} + 1451175927t^{3} + 156401037t^{2} + 1300925t + 15825 \right)D^{6} + 2 \left( t + 1 \right) ^{2} \left( 4180775819993136t^{12} + 7444630295586240t^{11} + 6500644393428721t^{10} + 3012726391963372t^{9} + 676867841493600t^{8} + 100710441811097t^{7} + 8724269489249t^{6} + 576551755725t^{5} + 43201691554t^{4} + 3248279350t^{3} + 308139699t^{2} + 1896000t + 15825 \right)D^{5} + \left( t + 1 \right) \left( 69679596999885600t^{13} + 169718187698416800t^{12} + 187170603999306534t^{11} + 117611519109300081t^{10} + 40194172086673371t^{9} + 7329125236252587t^{8} + 917121854013189t^{7} + 72507919342859t^{6} + 3372351629878t^{5} - 114198135106t^{4} - 6488923819t^{3} - 577650548t^{2} - 3233905t - 15825 \right)D^{4} + 2 t \left( 146327153699759760t^{13} + 463071689390389920t^{12} + 645130485844692815t^{11} + 518110384127302836t^{10} + 247827993994796583t^{9} + 66970642156963980t^{8} + 10493800734040815t^{7} + 1164866595280440t^{6} + 80419527108879t^{5} + 3374110031055t^{4} + 42278170590t^{3} + 430373126t^{2} + 416190t + 4275 \right)D^{3} + 4 t ^{2} \left( 161656665039734592t^{12} + 478058726471773584t^{11} + 630721410378528311t^{10} + 480762227197168527t^{9} + 213458257755025623t^{8} + 52192689377613882t^{7} + 7602726944550552t^{6} + 793228589089623t^{5} + 47621837126664t^{4} + 1686814877739t^{3} + 25831916355t^{2} + 335613557t + 288015 \right)D^{2} + 24 t ^{2} \left( 29265430739951952t^{12} + 82097458736702384t^{11} + 103874202873889014t^{10} + 75862488212343306t^{9} + 31505919900592965t^{8} + 7044150193824634t^{7} + 975259867601021t^{6} + 95979865423041t^{5} + 5037425736827t^{4} + 150113348803t^{3} + 2680726181t^{2} + 33805072t + 16880 \right)D + 144 t ^{3} \left( 1990845628568160t^{11} + 5380451437066720t^{10} + 6608482293883264t^{9} + 4673789942245728t^{8} + 1838390527126655t^{7} + 381497046544574t^{6} + 51012099399775t^{5} + 4736923507066t^{4} + 221745205417t^{3} + 5661417103t^{2} + 113746545t + 1281825 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 6)

|λ| = 10.47698 λ ≈ 10.47698; multiplicity 1; Jordan blocks 1 λ ≈ -2.800857 - 8.555346i; multiplicity 1; Jordan blocks 1 λ ≈ -2.800857 + 8.555346i; multiplicity 1; Jordan blocks 1 λ ≈ -2.437631 - 3.387688i; multiplicity 1; Jordan blocks 1 λ ≈ -2.437631 + 3.387688i; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 3; Jordan blocks 1, 2 3
Sector dimension
8
Spectral radius
$10.476977$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$10.47698$11
$-2.800857 - 8.555346\mathrm{i}$11
$-2.800857 + 8.555346\mathrm{i}$11
$-2.437631 - 3.387688\mathrm{i}$11
$-2.437631 + 3.387688\mathrm{i}$11
$-1$31, 2

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{3} \left(\lambda^{5} + 16 \lambda^{3} - 825 \lambda^{2} - 3750 \lambda - 14789\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda^{5} + 16 \lambda^{3} - 825 \lambda^{2} - 3750 \lambda - 14789\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 18 & 18 & 48 & 24 & 15 \\ 1 & -1 & 0 & 0 & 4 & 27 & 15 & 12 \\ 2 & 3 & 0 & 0 & -4 & 18 & 18 & 36 \\ 0 & 3 & 1 & -1 & 0 & 0 & 4 & 18 \\ 0 & 0 & 2 & 3 & 0 & 0 & -4 & 0 \\ 0 & 0 & 0 & 3 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 3 & \frac{5}{3} & 0\end{pmatrix}\notag\end{equation}$$