Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-124-32-a

·

This is a Fano fourfold of K3 type, appearing as case K3-27 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
Fl(1, 2, 8)
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_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(w7) + E^(w1 + w2) + 2 copies of E^(2w2)
Bundle
wedge^5 Q_1^v ot U_2^v ot U_3^v + O(1,1) + 2 copies of O(0,2)
ZL62 label
702.688w0
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,7) \\ \det\mathcal{E} &\cong \mathcal{O}(1,6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 32
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 124
$\chi(X, \mathrm{T}_X)$ = -28

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 124
$\mathrm{c}_1^2\mathrm{c}_2$ = 124
$\mathrm{c}_2^2$ = 124
$\mathrm{c}_1\mathrm{c}_3$ = 12
$\mathrm{c}_4$ = 36
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 28 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 + x2 + 28 0 x0 + 28 x2
$\mathrm{HH}^{ 3 }$ = x1 + 22 0 0 x1 22
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 29 0 0 x1 - x2 - 3 0 32
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 27t + 68t^{2} + 27t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(569)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(597)

Quantum differential equation

With $D=t\frac{\mathrm{d}}{\mathrm{d}t}$, the regularized period satisfies $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( t + 1 \right) ^{3} \left( 4528t^{3} + 288t^{2} - 16t - 1 \right) \left( 27154976340t^{6} + 6949518932t^{5} + 515309626t^{4} + 50110153t^{3} + 16556543t^{2} - 49801t + 775 \right)D^{6} + 2 \left( t + 1 \right) ^{2} \left( 799225263638880t^{10} + 501380267140352t^{9} + 105089577417680t^{8} + 11153942161622t^{7} + 1905974950140t^{6} + 379072307103t^{5} + 9109280876t^{4} + 71231385t^{3} + 34262297t^{2} - 89814t + 775 \right)D^{5} + \left( t + 1 \right) \left( 8268907535340720t^{11} + 9426953956713840t^{10} + 3182416014769976t^{9} + 497403131702204t^{8} + 57748123853348t^{7} + 13371964627928t^{6} + 1937098311419t^{5} + 13877298703t^{4} + 518544240t^{3} - 66705660t^{2} + 16537t - 775 \right)D^{4} + 2 t \left( 10912498791992400t^{11} + 19674015352270240t^{10} + 11432047745205504t^{9} + 2717260411416402t^{8} + 358229737579864t^{7} + 62953139093768t^{6} + 15100786768930t^{5} + 1440507717819t^{4} + 3538590260t^{3} + 120742410t^{2} + 2303906t + 225 \right)D^{3} + 4 t ^{2} \left( 7769391745566420t^{10} + 12201163153463416t^{9} + 5992980624255666t^{8} + 1228623974574812t^{7} + 152627641319944t^{6} + 35608974698208t^{5} + 8235522116117t^{4} + 550004922429t^{3} - 831644826t^{2} + 97060963t + 824507 \right)D^{2} + 16 t ^{2} \left( 1417856357128590t^{10} + 1996269687997836t^{9} + 854946774374679t^{8} + 158396126664269t^{7} + 20286308365377t^{6} + 5798458842634t^{5} + 1216946115577t^{4} + 53668228301t^{3} - 154367687t^{2} + 15123216t + 78120 \right)D + 32 t ^{2} \left( 207491174213940t^{10} + 268104123185040t^{9} + 102670765481994t^{8} + 17725336351514t^{7} + 2487153105505t^{6} + 804229430262t^{5} + 151718059810t^{4} + 4146347079t^{3} - 5536309t^{2} + 1762445t + 6200 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 8)

|λ| = 18.07223 λ ≈ -18.07223; multiplicity 1; Jordan blocks 1 λ ≈ 16.89877; multiplicity 1; Jordan blocks 1 λ ≈ -14.82653; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 3; Jordan blocks 1, 2 λ ≈ 0; multiplicity 2; Jordan blocks 2 3 2
Sector dimension
8
Spectral radius
$18.0722347$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$-18.07223$11
$16.89877$11
$-14.82653$11
$-1$31, 2
$0$22

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 1\right)^{3} \left(\lambda^{3} + 16 \lambda^{2} - 288 \lambda - 4528\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 1\right)^{2} \left(\lambda^{3} + 16 \lambda^{2} - 288 \lambda - 4528\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 16 & 24 & 12 & 448 & 320 & 320 \\ 1 & -1 & 4 & 16 & -32 & 48 & 264 & 192 \\ 1 & 2 & -4 & 32 & 72 & 0 & -216 & 256 \\ 0 & 2 & 1 & -1 & 8 & 32 & -24 & 0 \\ 0 & 0 & 1 & 2 & -8 & 16 & 56 & 24 \\ 0 & 0 & 0 & 2 & 1 & -1 & 4 & 16 \\ 0 & 0 & 0 & 0 & 1 & 2 & -4 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$