Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-224-51-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ4P4
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_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(2w3) + E^(w1 + w2 + w3)
Bundle
O_3(2) + O_1(1) box O_2(1) box O_3(1)
ZL62 label
1140.8x4000
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 51
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 224
$\chi(X, \mathrm{T}_X)$ = -3

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 224
$\mathrm{c}_1^2\mathrm{c}_2$ = 152
$\mathrm{c}_2^2$ = 100
$\mathrm{c}_1\mathrm{c}_3$ = 52
$\mathrm{c}_4$ = 16
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 8 0 0
0 0 0 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 + 3 0 x0 + 3 x1
$\mathrm{HH}^{ 3 }$ = x1 + 39 0 0 x1 - 6 45
$\mathrm{HH}^{ 4 }$ = 51 0 0 0 0 51
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 46t + 130t^{2} + 46t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(312)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(401)

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( 6t + 1 \right) \left( 2t - 1 \right) ^{2} \left( 2t + 1 \right) ^{2} \left( 464t^{4} + 432t^{3} + 32t^{2} + 4t - 1 \right) \left( 7410706560t^{11} - 37481110720t^{10} - 15060409632t^{9} + 24365066704t^{8} + 16624515208t^{7} + 4436586636t^{6} + 1006396602t^{5} + 210139263t^{4} + 11746825t^{3} + 1564581t^{2} + 17291t + 154 \right)D^{6} + \left( 2t - 1 \right) \left( 2t + 1 \right) \left( 1733038193295360t^{18} - 7725093796945920t^{17} - 11681592423161856t^{16} + 3093457375256576t^{15} + 10781729609491456t^{14} + 5606786265364992t^{13} + 1002431344194560t^{12} - 54114115285248t^{11} - 76428932425920t^{10} - 34817891375968t^{9} - 9594172315680t^{8} - 1474663201536t^{7} - 176206711236t^{6} - 18340120698t^{5} - 1747212224t^{4} - 58954925t^{3} - 6212482t^{2} - 50641t - 308 \right)D^{5} + \left( 57767939776512000t^{20} - 280496971503206400t^{19} - 359123748381409280t^{18} + 206176349184630784t^{17} + 411968350739664896t^{16} + 140895122287458304t^{15} - 27296592014575616t^{14} - 27658618335308288t^{13} - 8008306163657216t^{12} - 1831860047253888t^{11} - 224742029468480t^{10} + 68194062918368t^{9} + 25068198844752t^{8} + 3122440410888t^{7} + 346244390128t^{6} + 20114550914t^{5} - 1057717549t^{4} - 41655217t^{3} - 5602441t^{2} - 41765t - 154 \right)D^{4} + t \left( 242625347061350400t^{19} - 1253966968609996800t^{18} - 1349482484745584640t^{17} + 983562135285637120t^{16} + 1597480636417314816t^{15} + 541758486455027712t^{14} - 41128172382264320t^{13} - 68214483494435328t^{12} - 25133059899232000t^{11} - 7920550549392000t^{10} - 1308385696652992t^{9} + 74431609387872t^{8} + 42951118033904t^{7} + 6159516482840t^{6} + 875322942876t^{5} + 54184408026t^{4} + 1254850838t^{3} + 22053867t^{2} + 238974t + 99 \right)D^{3} + 2 t ^{2} \left( 268043240563015680t^{18} - 1449471339953111040t^{17} - 1349588153523920896t^{16} + 1182372911694573568t^{15} + 1658185295216090112t^{14} + 560233244928043520t^{13} + 2183887937977088t^{12} - 45419970649353088t^{11} - 21598334989623616t^{10} - 7984843301408800t^{9} - 1381235056706224t^{8} + 560743426680t^{7} + 18626239796296t^{6} + 3399388229772t^{5} + 541744721082t^{4} + 25921740351t^{3} + 459637478t^{2} + 10968072t + 100184 \right)D^{2} + 8 t ^{2} \left( 72787604118405120t^{18} - 406369512064573440t^{17} - 337403160201818112t^{16} + 339860410360031232t^{15} + 429392959087428352t^{14} + 144587577809145472t^{13} + 8169684173587520t^{12} - 7638845749881504t^{11} - 4825077172129040t^{10} - 1966148203449560t^{9} - 340783519967044t^{8} - 8916262820182t^{7} + 1622346643858t^{6} + 541371053131t^{5} + 91127073168t^{4} + 3386125920t^{3} + 52776454t^{2} + 1359214t + 10780 \right)D + 48 t ^{2} \left( 4951537695129600t^{18} - 28230762171187200t^{17} - 21589011016212480t^{16} + 23978900296261632t^{15} + 28252308013510400t^{14} + 9484629537348224t^{13} + 853379382163264t^{12} - 336372115264032t^{11} - 286406460117328t^{10} - 124085922397336t^{9} - 21304395511460t^{8} - 877961194982t^{7} + 2765762978t^{6} + 25180298489t^{5} + 4250976486t^{4} + 127211403t^{3} + 2038784t^{2} + 46607t + 308 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ4P4

|λ| = 10.91482 λ ≈ 10.91482; multiplicity 1; Jordan blocks 1 λ ≈ -2.880191 - 5.341209i; multiplicity 1; Jordan blocks 1 λ ≈ -2.880191 + 5.341209i; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2; multiplicity 3; Jordan blocks 1, 2 λ ≈ 2; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ -1.154437; multiplicity 1; Jordan blocks 1 2 3 3
Sector dimension
12
Spectral radius
$10.9148198$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$10.91482$11
$-2.880191 - 5.341209\mathrm{i}$11
$-2.880191 + 5.341209\mathrm{i}$11
$-6$21, 1
$-2$31, 2
$2$31, 1, 1
$-1.154437$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda + 6\right)^{2} \left(\lambda - 2\right)^{3} \left(\lambda^{4} - 4 \lambda^{3} - 32 \lambda^{2} - 432 \lambda - 464\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 6\right) \left(\lambda - 2\right) \left(\lambda^{4} - 4 \lambda^{3} - 32 \lambda^{2} - 432 \lambda - 464\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 2 & 4 & 0 & 6 & 6 & 12 & 16 & 16 & 16 & 0 \\ 1 & -1 & 1 & 0 & 0 & 4 & 0 & 4 & 6 & 6 & 6 & 0 \\ 1 & 1 & -1 & 0 & 0 & 0 & 4 & 4 & 6 & 6 & 6 & 0 \\ 2 & 1 & 1 & 0 & 2 & 2 & 2 & 4 & 0 & 6 & 6 & 16 \\ 0 & 1 & 1 & 0 & -2 & 0 & 0 & 0 & 4 & 4 & 4 & 0 \\ 0 & 2 & 0 & 1 & 1 & -1 & 1 & 0 & 0 & 4 & 0 & 6 \\ 0 & 0 & 2 & 1 & 1 & 1 & -1 & 0 & 0 & 0 & 4 & 6 \\ 0 & 0 & 0 & 2 & 0 & 1 & 1 & 0 & 2 & 2 & 2 & 0 \\ 0 & 0 & 0 & 0 & 2 & 1 & 1 & -4 & -2 & -2 & -2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 3 & 1 & 0 & 2 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 1 & 2 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 3 & 0\end{pmatrix}\notag\end{equation}$$