Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-112-30-g

·

No geometric description is recorded for this family (yet).

Defining data
Ambient
ℙ1P1 × ℙ4P4
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^(w1 + 3w2)
Bundle
O_1(1) box O_2(3)
ZL62 label
140.x302
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 30
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 112
$\chi(X, \mathrm{T}_X)$ = -42

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 112
$\mathrm{c}_1^2\mathrm{c}_2$ = 124
$\mathrm{c}_2^2$ = 136
$\mathrm{c}_1\mathrm{c}_3$ = -4
$\mathrm{c}_4$ = 68
Hodge diamond
1
00
0 2 0
0 0 0 0
0 5 52 5 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = 42 0 42 0
$\mathrm{HH}^{ 3 }$ = 20 0 0 0 20
$\mathrm{HH}^{ 4 }$ = 45 0 0 15 0 30
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 25t + 60t^{2} + 25t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(528)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(566)

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( 8539t^{5} + 6039t^{4} - 2700t^{3} - 216t^{2} + 1 \right) \left( 1091829624t^{9} + 36015207328t^{8} + 90633887761t^{7} + 85694840802t^{6} + 36943300846t^{5} + 6290119824t^{4} - 547218268t^{3} - 34257672t^{2} - 281356t - 2800 \right)D^{6} + \left( t + 1 \right) ^{2} \left( 195785796346056t^{15} + 6961743521346160t^{14} + 24785131850997573t^{13} + 36413060982237038t^{12} + 26906799388661671t^{11} + 9686723208997240t^{10} + 645708058189022t^{9} - 682687250822605t^{8} - 147368069878481t^{7} + 8180530301742t^{6} + 985089745226t^{5} - 4172936980t^{4} + 2852380844t^{3} + 138127284t^{2} + 816068t + 5600 \right)D^{5} + \left( t + 1 \right) \left( 1631548302883800t^{16} + 60774624454531240t^{15} + 246795118320731261t^{14} + 433960396624155061t^{13} + 412929004262694277t^{12} + 224533400674857263t^{11} + 63895964683018688t^{10} + 4041648707222752t^{9} - 2252102669477110t^{8} - 278629989764559t^{7} + 35278168826330t^{6} + 3429243102114t^{5} + 143478513760t^{4} - 1664128500t^{3} - 133938752t^{2} - 900292t - 2800 \right)D^{4} + t \left( 6852502872111960t^{16} + 265830130044560320t^{15} + 1232498930545435375t^{14} + 2553055949847723312t^{13} + 2972224668158027000t^{12} + 2099492145474037456t^{11} + 895894272227552603t^{10} + 201935699830870996t^{9} + 8657996307125981t^{8} - 4520633586286487t^{7} - 291614531694945t^{6} + 47599815458774t^{5} + 4291076981210t^{4} + 93807520380t^{3} + 1613454912t^{2} + 12606420t + 2700 \right)D^{3} + t ^{2} \left( 15140768250761664t^{15} + 593653680596138464t^{14} + 2617540921419458672t^{13} + 5129787206385319176t^{12} + 5639912273178634046t^{11} + 3756939489232598552t^{10} + 1502979824834280030t^{9} + 310594395385859536t^{8} + 9926121925085419t^{7} - 5841813213361254t^{6} - 280279008977196t^{5} + 24837336652624t^{4} + 3662388531528t^{3} + 84439531872t^{2} + 1628096148t + 9477216 \right)D^{2} + 4 t ^{2} \left( 4111501723267176t^{15} + 162460513395363472t^{14} + 689011413013569489t^{13} + 1292306341623171072t^{12} + 1355987713900807145t^{11} + 858896817442014510t^{10} + 323142588260172124t^{9} + 60379332424959760t^{8} + 755524825515420t^{7} - 1123751013394000t^{6} - 50241809210172t^{5} + 860172544320t^{4} + 454767773056t^{3} + 11378332800t^{2} + 209662176t + 900480 \right)D + 48 t ^{2} \left( 139846997390040t^{15} + 5554657261928080t^{14} + 22924572933072303t^{13} + 41658019509300180t^{12} + 42218480335150509t^{11} + 25714381406815458t^{10} + 9188849160033832t^{9} + 1559171611206456t^{8} - 12253732632678t^{7} - 29989067956932t^{6} - 1393009077980t^{5} - 6839463696t^{4} + 8410492032t^{3} + 225690432t^{2} + 3666120t + 11200 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ4P4

|λ| = 18.51642 λ ≈ 18.51642; multiplicity 1; Jordan blocks 1 λ ≈ -10.147 - 7.9217i; multiplicity 1; Jordan blocks 1 λ ≈ -10.147 + 7.9217i; multiplicity 1; Jordan blocks 1 λ ≈ 2.778979; multiplicity 1; Jordan blocks 1 λ ≈ -1.001389; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 3; Jordan blocks 3 3
Sector dimension
8
Spectral radius
$18.516419$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$18.51642$11
$-10.147 - 7.9217\mathrm{i}$11
$-10.147 + 7.9217\mathrm{i}$11
$2.778979$11
$-1.001389$11
$-1$33

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{3} \left(\lambda^{5} - 216 \lambda^{3} - 2700 \lambda^{2} + 6039 \lambda + 8539\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right)^{3} \left(\lambda^{5} - 216 \lambda^{3} - 2700 \lambda^{2} + 6039 \lambda + 8539\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 12 & 54 & 54 & 72 & 108 & 555 \\ 1 & -1 & 0 & 12 & 22 & 45 & 24 & 84 \\ 2 & 3 & 0 & 0 & 30 & 135 & 90 & 72 \\ 0 & 2 & 1 & -1 & 0 & 30 & 28 & 36 \\ 0 & 0 & 2 & 3 & 0 & 0 & 12 & 54 \\ 0 & 0 & 0 & 2 & 1 & -1 & 0 & 12 \\ 0 & 0 & 0 & 0 & 2 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & \frac{5}{3} & 0\end{pmatrix}\notag\end{equation}$$