Family 2-85-24-a

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

Fl(1, 3, 5) · 44.3TT (preferred model)

Defining data

Ambient: Fl(1, 3, 5)
Bundle: $E^{ω_{4}} \oplus (E^{ω_{1} + ω_{3}})^{\oplus 2}$ E^(w4) + 2 copies of E^(w1 + w3)
Bundle: $Q_{1}^{\lor} \otimes ⋀^{2} U_{2}^{\lor} \otimes U_{3}^{\lor} \oplus O (1, 1)^{\oplus 2}$ Q_1^v ot wedge^2 U_2^v ot U_3^v + 2 copies of O(1,1)
ZL62 label: 44.3TT

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,4) \\ \det\mathcal{E} &\cong \mathcal{O}(2,3) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	24
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	85
$\chi(X, \mathrm{T}_X)$	=	-28

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	85
$\mathrm{c}_1^2\mathrm{c}_2$	=	106
$\mathrm{c}_2^2$	=	137
$\mathrm{c}_1\mathrm{c}_3$	=	12
$\mathrm{c}_4$	=	42

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		2		32		2		0
	0		0		0		0
		0		2		0
			0		0
				1

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = x₀	0	x₀
$\mathrm{HH}^{ 2 }$ = x₀ + x₁ + 28	0	x₀ + 28	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 14	0	0	x₁ + 8	6
$\mathrm{HH}^{ 4 }$ = 27	0	0	3	0	24
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 19t + 45t^{2} + 19t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(625)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	No match

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) ^{4} \left( 20423t^{6} + 31470t^{5} + 19678t^{4} + 5200t^{3} + 421t^{2} - 2t - 1 \right) \left( 255375003319101t^{13} + 781529546856096t^{12} + 1072634400920898t^{11} + 868854628692753t^{10} + 457920611620353t^{9} + 162698231732928t^{8} + 39150922047491t^{7} + 6380232053231t^{6} + 753548142390t^{5} + 83491359295t^{4} + 9501760960t^{3} + 615211200t^{2} - 128100t + 20400 \right)D^{6} + \left( 2t + 1 \right) ^{3} \left( 219051995097011988366t^{20} + 1022477688445895824239t^{19} + 2223557461569379308462t^{18} + 2982539833189676485848t^{17} + 2752893108985871560104t^{16} + 1846569493602053323296t^{15} + 926644334474363286936t^{14} + 352994305052033141198t^{13} + 102748675325035876364t^{12} + 22983822108867577219t^{11} + 4024893226950247640t^{10} + 582329211014266712t^{9} + 75900522152119572t^{8} + 9056485792380809t^{7} + 853949630716278t^{6} + 50042147332010t^{5} + 1395871400850t^{4} + 38772418700t^{3} + 2513794800t^{2} - 873900t + 40800 \right)D^{5} + \left( 2t + 1 \right) ^{2} \left( 3650866584950199806100t^{21} + 17600632909531166744100t^{20} + 39572153670781998491715t^{19} + 55010236010920455728946t^{18} + 52819276113955155961200t^{17} + 37043805005894943012261t^{16} + 19562565880888026880426t^{15} + 7906396422568139717270t^{14} + 2466225305384615617246t^{13} + 597977571561181414168t^{12} + 114639741900501344871t^{11} + 18177341042094207694t^{10} + 2571213303037842506t^{9} + 335928182520483749t^{8} + 36803915146007995t^{7} + 2792719798790658t^{6} + 112760392673900t^{5} + 1143324861045t^{4} - 32097285340t^{3} - 2437188900t^{2} - 2691000t - 20400 \right)D^{4} + t \left( 2t + 1 \right) \left( 30667279313581678371240t^{21} + 154822727840374140320700t^{20} + 365259332867750291940750t^{19} + 534328966415049707050353t^{18} + 541952220592267750291410t^{17} + 403457788574985029760168t^{16} + 227555882433842363766600t^{15} + 98981186301499277400920t^{14} + 33539898901624029084784t^{13} + 8925725474244870624282t^{12} + 1894043917728533620600t^{11} + 332431477732050519713t^{10} + 51492821069276274946t^{9} + 7388091117993448824t^{8} + 933802447996569408t^{7} + 90238449051340355t^{6} + 5582308514331984t^{5} + 170397721208370t^{4} + 1297412142900t^{3} + 16195469220t^{2} + 224790300t + 50400 \right)D^{3} + 2 t ^{2} \left( 67760083816675708401216t^{21} + 361838540417308015061808t^{20} + 905024067571652532778260t^{19} + 1407632009415036574141032t^{18} + 1523415081073837629688707t^{17} + 1215581189381133894491850t^{16} + 738999957861439718163272t^{15} + 348887543568372997906146t^{14} + 129362519281077973543478t^{13} + 37991890660522550147404t^{12} + 8949218317705306221687t^{11} + 1738153440638069266598t^{10} + 293178913465276628843t^{9} + 45355879616289868074t^{8} + 6378108942454424965t^{7} + 739034985173617640t^{6} + 61383093797562066t^{5} + 3093444764193718t^{4} + 71700802844870t^{3} + 454190792310t^{2} + 12023507940t + 76739700 \right)D^{2} + 4 t ^{2} \left( 36800735176298014045488t^{21} + 190893641497654907996784t^{20} + 463385336465427928282260t^{19} + 698765730516564675591672t^{18} + 732253937384474512502859t^{17} + 564805500722818797202302t^{16} + 331210542942637240878822t^{15} + 150459181880514667948600t^{14} + 53559495714098263656400t^{13} + 15093120117786885039396t^{12} + 3426876525795956740531t^{11} + 651020206716322263874t^{10} + 110033295050611050903t^{9} + 17277827103035662908t^{8} + 2415773258961018789t^{7} + 267221727558210890t^{6} + 20319371751913280t^{5} + 893183508823340t^{4} + 16706249316470t^{3} + 119148968550t^{2} + 3296229600t + 13137600 \right)D + 8 t ^{2} \left( 7510354117611839601120t^{21} + 38182725699291493328160t^{20} + 90768446192783097065880t^{19} + 133915633653576948791184t^{18} + 137137274036486290033659t^{17} + 103210349210248104713736t^{16} + 58941531070852990663134t^{15} + 26017508171527483818516t^{14} + 8981483155672021970091t^{13} + 2453662597984314324348t^{12} + 542531638643749975016t^{11} + 101738533957971902554t^{10} + 17300368198425803042t^{9} + 2744712158768089150t^{8} + 379297094448148149t^{7} + 40137426695469996t^{6} + 2824359907516915t^{5} + 110096683247220t^{4} + 1704993753950t^{3} + 16393017750t^{2} + 343278000t + 897600 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 5)

Show multiplicities

Sector dimension: 10
Spectral radius: $24.8023735$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$24.80237$	1	1
$-10.7992 - 2.292732\mathrm{i}$	1	1
$-10.7992 + 2.292732\mathrm{i}$	1	1
$-2.472936$	1	1
$-2$	4	2, 2
$-1.365515 - 0.9313342\mathrm{i}$	1	1
$-1.365515 + 0.9313342\mathrm{i}$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{4} \left(\lambda^{6} + 2 \lambda^{5} - 421 \lambda^{4} - 5200 \lambda^{3} - 19678 \lambda^{2} - 31470 \lambda - 20423\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda^{6} + 2 \lambda^{5} - 421 \lambda^{4} - 5200 \lambda^{3} - 19678 \lambda^{2} - 31470 \lambda - 20423\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 22 & 22 & 186 & 126 & 78 & 96 & 776 & 672 & 1920 \\ 1 & -1 & 4 & 32 & 30 & 8 & 10 & 141 & 117 & 312 \\ 1 & 3 & -1 & 42 & 26 & 18 & 22 & 114 & 99 & 360 \\ 0 & 1 & 1 & -2 & 0 & 1 & 1 & 20 & 18 & 36 \\ 0 & 1 & 0 & 4 & -1 & 2 & 3 & 6 & 6 & 36 \\ 0 & 0 & 1 & 3 & 3 & -2 & -1 & 8 & 6 & 18 \\ 0 & 0 & 1 & 3 & 4 & 0 & -1 & 22 & 18 & 36 \\ 0 & 0 & 0 & -2 & -6 & 2 & 3 & -8 & -3 & 0 \\ 0 & 0 & 0 & \frac{19}{3} & \frac{29}{3} & -\frac{2}{3} & -\frac{5}{3} & \frac{37}{3} & 6 & 22 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{3} & 2 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta