Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-160-39-a

·

This is a Fano fourfold of K3 type, appearing as case K3-41 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ3P3
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_5ω1ω2ω3ω4ω5
Bundle
E^(w1 + w2 + 2w3)
Bundle
O_1(1) box O_2(1) box O_3(2)
ZL62 label
1130.x4001
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;4) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 39
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 160
$\chi(X, \mathrm{T}_X)$ = -18

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 160
$\mathrm{c}_1^2\mathrm{c}_2$ = 136
$\mathrm{c}_2^2$ = 112
$\mathrm{c}_1\mathrm{c}_3$ = 32
$\mathrm{c}_4$ = 32
Hodge diamond
1
00
0 3 0
0 0 0 0
0 1 22 1 0
0 0 0 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = x0 + 18 0 18 x0
$\mathrm{HH}^{ 3 }$ = x0 + 27 0 0 x0 + 1 26
$\mathrm{HH}^{ 4 }$ = 40 0 0 1 0 39
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 34t + 90t^{2} + 34t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(355)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(432)

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( 2t - 1 \right) \left( 6t + 1 \right) \left( 14t - 1 \right) \left( 2t + 1 \right) ^{4} \left( 68t^{2} + 4t + 1 \right) \left( 656880512t^{9} + 1304640960t^{8} + 1552331456t^{7} + 1061661408t^{6} + 357785400t^{5} + 51461852t^{4} + 2821360t^{3} + 185976t^{2} + 858t + 15 \right)D^{6} + 2 \left( 2t + 1 \right) ^{3} \left( 157588262350848t^{15} + 317529856237568t^{14} + 376407441394688t^{13} + 241775167331328t^{12} + 49788056998400t^{11} - 20429330063360t^{10} - 12806128274304t^{9} - 2615192327808t^{8} - 270042482256t^{7} - 21140744032t^{6} - 1728608868t^{5} - 158724452t^{4} - 5787916t^{3} - 369864t^{2} - 1227t - 15 \right)D^{5} + \left( 2t + 1 \right) ^{2} \left( 5252942078361600t^{16} + 12684478717665280t^{15} + 17390522957512704t^{14} + 14219838907697152t^{13} + 6002484456954880t^{12} + 782069126004736t^{11} - 299288403212288t^{10} - 133492243069696t^{9} - 21489627621952t^{8} - 1931070560352t^{7} - 127326730560t^{6} - 2111972888t^{5} + 319670820t^{4} + 12069916t^{3} + 685608t^{2} + 2304t + 15 \right)D^{4} + 2 t \left( 2t + 1 \right) \left( 22062356729118720t^{16} + 62569130131619840t^{15} + 97275436406620160t^{14} + 94367119910445056t^{13} + 54411496378012672t^{12} + 16883428066191360t^{11} + 1801572480821248t^{10} - 437841582062080t^{9} - 167756725356608t^{8} - 23587093822208t^{7} - 2108598029184t^{6} - 144619137216t^{5} - 4713326260t^{4} - 65913984t^{3} - 901620t^{2} - 13524t - 9 \right)D^{3} + 4 t ^{2} \left( 48747302487195648t^{16} + 159685515483308032t^{15} + 278461180405022720t^{14} + 309868784449130496t^{13} + 219652315594690560t^{12} + 95951014682874368t^{11} + 23805726577616640t^{10} + 2361741971205632t^{9} - 290142061987264t^{8} - 112606007463264t^{7} - 15052741720880t^{6} - 1383634430272t^{5} - 86514061908t^{4} - 2193483770t^{3} - 30430284t^{2} - 657045t - 5430 \right)D^{2} + 16 t ^{2} \left( 13237414037471232t^{16} + 42778592832942080t^{15} + 74494022481485824t^{14} + 82813415653174272t^{13} + 58410512166269440t^{12} + 25433765299520256t^{11} + 6446230735301952t^{10} + 777959659882208t^{9} - 11136260045824t^{8} - 15494022029568t^{7} - 2256327858316t^{6} - 226528771398t^{5} - 13331620296t^{4} - 267457471t^{3} - 4007628t^{2} - 89550t - 555 \right)D + 96 t ^{2} \left( 900504356290560t^{16} + 2883255250739200t^{15} + 5015771035910144t^{14} + 5569167700945920t^{13} + 3909155247829504t^{12} + 1692722608178176t^{11} + 431117380963008t^{10} + 55955356973536t^{9} + 1284344575616t^{8} - 583895476416t^{7} - 97395135348t^{6} - 10709339022t^{5} - 581934528t^{4} - 9552529t^{3} - 164520t^{2} - 3069t - 15 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ3P3

|λ| = 14 λ ≈ 14; multiplicity 1; Jordan blocks 1 λ ≈ -2 - 8i; multiplicity 1; Jordan blocks 1 λ ≈ -2 + 8i; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2; multiplicity 5; Jordan blocks 1, 2, 2 λ ≈ 2; multiplicity 2; Jordan blocks 1, 1 2 5 2
Sector dimension
12
Spectral radius
$14$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$14$11
$-2 - 8\mathrm{i}$11
$-2 + 8\mathrm{i}$11
$-6$21, 1
$-2$51, 2, 2
$2$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{5} \left(\lambda + 6\right)^{2} \left(\lambda - 14\right) \left(\lambda - 2\right)^{2} \left(\lambda^{2} + 4 \lambda + 68\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 6\right) \left(\lambda - 14\right) \left(\lambda - 2\right) \left(\lambda^{2} + 4 \lambda + 68\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 2 & 4 & 0 & 12 & 12 & 24 & 48 & 56 & 56 & 80 \\ 1 & -1 & 1 & 0 & 0 & 4 & 0 & 6 & 12 & 18 & 18 & 32 \\ 1 & 1 & -1 & 0 & 0 & 0 & 4 & 6 & 12 & 18 & 18 & 32 \\ 2 & 2 & 2 & 0 & 4 & 2 & 2 & 4 & 0 & 12 & 12 & 48 \\ 0 & 1 & 1 & 0 & -2 & 0 & 0 & 0 & 4 & 6 & 6 & 12 \\ 0 & 2 & 0 & 1 & 2 & -1 & 1 & 0 & 0 & 4 & 0 & 12 \\ 0 & 0 & 2 & 1 & 2 & 1 & -1 & 0 & 0 & 0 & 4 & 12 \\ 0 & 0 & 0 & 2 & 0 & 2 & 2 & 0 & 4 & 2 & 2 & 0 \\ 0 & 0 & 0 & 0 & 2 & 1 & 1 & -1 & -2 & -1 & -1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 2 & 0 & 2 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 2 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$