Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-144-36-b

·

This is a Fano fourfold of K3 type, appearing as case C-1 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ1P1 × ℙ5P5
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_6ω1ω2ω3ω4ω5ω6
Bundle
E^(3w2) + E^(w1 + w2)
Bundle
O_2(3) + O_1(1) box O_2(1)
ZL62 label
150.Dg
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;6) \\ \det\mathcal{E} &\cong \mathcal{O}(1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 36
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 144
$\chi(X, \mathrm{T}_X)$ = -28

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 144
$\mathrm{c}_1^2\mathrm{c}_2$ = 132
$\mathrm{c}_2^2$ = 120
$\mathrm{c}_1\mathrm{c}_3$ = 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 + 32 0 x0 + 28 4
$\mathrm{HH}^{ 3 }$ = x1 + 30 0 0 x1 30
$\mathrm{HH}^{ 4 }$ = x1 + 33 0 0 x1 - 3 0 36
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 31t + 80t^{2} + 31t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(502)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(551)

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) ^{2} \left( 10935t^{4} - 972t^{3} - 216t^{2} + t + 1 \right) \left( 52612659000t^{9} + 105863047200t^{8} + 9436850325t^{7} - 16086966930t^{6} - 8435178486t^{5} - 3470270616t^{4} + 116425860t^{3} - 175160t^{2} - 23388t + 144 \right)D^{6} + \left( t + 1 \right) \left( 7479152540145000t^{14} + 18622915485741000t^{13} + 7356603906298125t^{12} - 2501981486857575t^{11} - 2784074926764150t^{10} - 1315629107824419t^{9} - 251848075537098t^{8} + 41762129677713t^{7} + 529480869768t^{6} - 16670283726t^{5} + 21494209640t^{4} - 559616528t^{3} + 375128t^{2} + 71316t - 288 \right)D^{5} + \left( 38546401553055000t^{15} + 122574418832754000t^{14} + 95719147621901325t^{13} + 2378971137069900t^{12} - 27093891815936775t^{11} - 19549407049642656t^{10} - 7422148449511326t^{9} - 896138794527993t^{8} + 99732336528051t^{7} - 4075407553332t^{6} - 130255502082t^{5} - 29398631084t^{4} + 678054924t^{3} - 297892t^{2} - 33060t + 144 \right)D^{4} + t \left( 100680899578875000t^{14} + 298122390808434000t^{13} + 177489304837186275t^{12} - 28463585551061850t^{11} - 73603269884110275t^{10} - 49321740508338960t^{9} - 17460909702691947t^{8} - 1679361702053175t^{7} + 126504457523307t^{6} - 7140548505342t^{5} + 64958722462t^{4} + 1701714060t^{3} + 16011696t^{2} - 569468t - 36 \right)D^{3} + t ^{2} \left( 140377939984260000t^{13} + 394169353918380000t^{12} + 174157219948502700t^{11} - 66874609869824700t^{10} - 106839031661194950t^{9} - 67637601382196952t^{8} - 21936560002340193t^{7} - 1484855017492878t^{6} + 89608734444276t^{5} - 4116730780200t^{4} + 101959006376t^{3} + 1163574800t^{2} - 11308908t - 420384 \right)D^{2} + 12 t ^{2} \left( 8246245108365000t^{13} + 22275128435544000t^{12} + 7119321770801925t^{11} - 4840887486227850t^{10} - 6496457558680800t^{9} - 3953889048616488t^{8} - 1180547013378636t^{7} - 47901260256048t^{6} + 2749706935956t^{5} + 6440054528t^{4} + 3640331328t^{3} + 37859296t^{2} - 1325136t - 13824 \right)D + 144 t ^{2} \left( 191773142055000t^{13} + 503646635700000t^{12} + 112827249354375t^{11} - 125558383520250t^{10} - 155598038806050t^{9} - 92262644641656t^{8} - 25722649944846t^{7} - 450966681252t^{6} + 34383550932t^{5} + 3739285616t^{4} + 31505512t^{3} + 522704t^{2} - 34536t - 192 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ5P5

|λ| = 14.72597 λ ≈ 14.72597; multiplicity 1; Jordan blocks 1 λ ≈ -10.6393 - 4.531849i; multiplicity 1; Jordan blocks 1 λ ≈ -10.6393 + 4.531849i; multiplicity 1; Jordan blocks 1 λ ≈ 5.552625; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 2; Jordan blocks 2 λ ≈ 0; multiplicity 2; Jordan blocks 2 2 2
Sector dimension
8
Spectral radius
$14.7259732$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$14.72597$11
$-10.6393 - 4.531849\mathrm{i}$11
$-10.6393 + 4.531849\mathrm{i}$11
$5.552625$11
$-1$22
$0$22

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 1\right)^{2} \left(\lambda^{4} + \lambda^{3} - 216 \lambda^{2} - 972 \lambda + 10935\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 1\right)^{2} \left(\lambda^{4} + \lambda^{3} - 216 \lambda^{2} - 972 \lambda + 10935\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 12 & 18 & 18 & 0 & 72 & 180 \\ 1 & -1 & 0 & 12 & 12 & 0 & 0 & 72 \\ 2 & 1 & 0 & 0 & 30 & 45 & 45 & 0 \\ 0 & 2 & 1 & -1 & 0 & 30 & 30 & 0 \\ 0 & 0 & 2 & 1 & 0 & 0 & 12 & 18 \\ 0 & 0 & 0 & 2 & 1 & -1 & 0 & 12 \\ 0 & 0 & 0 & 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 3 & 0\end{pmatrix}\notag\end{equation}$$