Family 3-288-63-b

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

Fl(1, 4, 7) · 608.55596 (preferred model)

Defining data

Ambient: Fl(1, 4, 7)
Bundle: $(E^{ω_{6}})^{\oplus 3} \oplus E^{2 ω_{1}} \oplus E^{2 ω_{4}}$ 3 copies of E^(w6) + E^(2w1) + E^(2w4)
Bundle: $(⋀^{2} Q_{1}^{\lor} \otimes ⋀^{3} U_{2}^{\lor} \otimes U_{3}^{\lor})^{\oplus 3} \oplus O (2, 0) \oplus O (0, 2)$ 3 copies of wedge^2 Q_1^v ot wedge^3 U_2^v ot U_3^v + O(2,0) + O(0,2)
ZL62 label: 608.55596

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4,6) \\ \det\mathcal{E} &\cong \mathcal{O}(2,5) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	63
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	288
$\chi(X, \mathrm{T}_X)$	=	1

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	288
$\mathrm{c}_1^2\mathrm{c}_2$	=	168
$\mathrm{c}_2^2$	=	100
$\mathrm{c}_1\mathrm{c}_3$	=	52
$\mathrm{c}_4$	=	16

Hodge diamond

				1
			0		0
		0		3		0
	0		0		0		0
0		0		8		0		0
	0		0		0		0
		0		3		0
			0		0
				1

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = x₀	0	x₀
$\mathrm{HH}^{ 2 }$ = x₀ + x₂ - 1	0	x₀ - 1	x₂
$\mathrm{HH}^{ 3 }$ = x₁ + 73	0	0	x₁	73
$\mathrm{HH}^{ 4 }$ = x₁ - x₂ + 89	0	0	x₁ - x₂ + 26	0	63
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$

$\displaystyle\frac{ 1 + 58t + 170t^{2} + 58t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(221)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(299)

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( 4t - 1 \right) \left( 4t + 1 \right) ^{2} \left( 1472t^{4} + 576t^{3} + 32t^{2} - 4t - 1 \right) \left( 9763528704t^{11} + 46618828800t^{10} + 70283275776t^{9} + 44654040320t^{8} + 14273860640t^{7} + 1980749024t^{6} - 119544224t^{5} - 78317311t^{4} - 8999965t^{3} - 308065t^{2} + 2427t - 126 \right)D^{6} + \left( 4t + 1 \right) \left( 2989358164475904t^{17} + 16066383993372672t^{16} + 28513972150861824t^{15} + 22780854109732864t^{14} + 9522745699270656t^{13} + 1901204540080128t^{12} - 33762913323008t^{11} - 104885062428672t^{10} - 20249855346176t^{9} - 620525763840t^{8} + 273373394336t^{7} + 40904665888t^{6} + 3767666416t^{5} + 480286022t^{4} + 42807609t^{3} + 1297672t^{2} - 9297t + 252 \right)D^{5} + \left( 61856718941847552t^{18} + 357624946517606400t^{17} + 703105089293254656t^{16} + 648328812249350144t^{15} + 333883588922736640t^{14} + 98301103554289664t^{13} + 12866195275415552t^{12} - 1324593907345920t^{11} - 777040506221056t^{10} - 101074774205568t^{9} + 1751913816176t^{8} + 1902787185792t^{7} + 209228202396t^{6} + 7078086298t^{5} - 423269847t^{4} - 48004947t^{3} - 1421099t^{2} - 207t - 126 \right)D^{4} + t \left( 163264945905991680t^{17} + 963407151265480704t^{16} + 1921892142120173568t^{15} + 1775742561283997696t^{14} + 918100315349385216t^{13} + 272495273146318848t^{12} + 35999470590496768t^{11} - 3835918634902528t^{10} - 2343748019740160t^{9} - 368646944923072t^{8} - 19546747212640t^{7} + 1234148246208t^{6} + 203487935744t^{5} + 7591703684t^{4} + 85554956t^{3} + 7554227t^{2} + 130580t + 21 \right)D^{3} + 2 t ^{2} \left( 116240042472505344t^{16} + 696627992482283520t^{15} + 1407381266759221248t^{14} + 1303049024048922624t^{13} + 674977914528313344t^{12} + 199618922627209216t^{11} + 24984009170106624t^{10} - 3699205962551296t^{9} - 2001654403127296t^{8} - 331111951548520t^{7} - 24026214561304t^{6} - 183441709080t^{5} + 73081995008t^{4} + 3026094057t^{3} + 54329546t^{2} + 4925394t + 51696 \right)D^{2} + 16 t ^{2} \left( 10606472718188544t^{16} + 64327499616878592t^{15} + 131357174683852800t^{14} + 121877866559324160t^{13} + 63236629590829568t^{12} + 18576514836597504t^{11} + 2150229086702176t^{10} - 448991990777760t^{9} - 216946859017472t^{8} - 36413030997027t^{7} - 2939988543118t^{6} - 78471980595t^{5} + 3268365260t^{4} + 184036312t^{3} + 5545694t^{2} + 400590t + 2772 \right)D + 96 t ^{2} \left( 517388913082368t^{16} + 3167076748099584t^{15} + 6524701334495232t^{14} + 6065934344306688t^{13} + 3152622021749248t^{12} + 919694164894464t^{11} + 97642479900512t^{10} - 27456548296608t^{9} - 12234182332608t^{8} - 2058779501843t^{7} - 173869161242t^{6} - 5900518195t^{5} + 68696136t^{4} + 7138400t^{3} + 318800t^{2} + 17605t + 84 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 4, 7)

Show multiplicities

Sector dimension: 9
Spectral radius: $8.9148198$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$8.91482$	1	1
$-4.880191 - 5.341209\mathrm{i}$	1	1
$-4.880191 + 5.341209\mathrm{i}$	1	1
$-4$	2	1, 1
$4$	1	1
$-3.154437$	1	1
$0$	2	2

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 4\right)^{2} \left(\lambda - 4\right) \left(\lambda^{4} + 4 \lambda^{3} - 32 \lambda^{2} - 576 \lambda - 1472\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 4\right) \left(\lambda - 4\right) \left(\lambda^{4} + 4 \lambda^{3} - 32 \lambda^{2} - 576 \lambda - 1472\right)\notag\end{equation}$$

Quantum multiplication matrix

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

Fano fourfolds of type A beta