Family 2-99-27-b

This is a Fano fourfold of K3 type, appearing as case R-61 in Bernardara–Fatighenti–Manivel–Tanturri.

ℙ²P2 × Gr(2, 6) · 2051.wBw6 (preferred model)

Defining data

Ambient: ℙ²P2 × Gr(2, 6)
Bundle: $E^{ω_{3} + ω_{4}} \oplus E^{ω_{2} + ω_{3}}$ E^(w3 + w4) + E^(w2 + w3)
Bundle: $U_{2}^{\lor} (1) \oplus Q_{1} ⊠ U_{2}^{\lor}$ U_2^v(1) + Q_1 box U_2^v
ZL62 label: 2051.wBw6

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;6) \\ \det\mathcal{E} &\cong \mathcal{O}(2;5) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	27
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	99
$\chi(X, \mathrm{T}_X)$	=	-25

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	99
$\mathrm{c}_1^2\mathrm{c}_2$	=	114
$\mathrm{c}_2^2$	=	126
$\mathrm{c}_1\mathrm{c}_3$	=	18
$\mathrm{c}_4$	=	33

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		1		25		1		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 + x₂ + 25	0	x₀ + 25	x₂
$\mathrm{HH}^{ 3 }$ = x₁ + x₃	0	0	x₁	x₃
$\mathrm{HH}^{ 4 }$ = x₁ - x₂ + x₃ + 7	0	0	x₁ - x₂ - 11	x₃ - 9	27
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 22t + 53t^{2} + 22t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(562)
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) \left( 5t + 1 \right) \left( 22t - 1 \right) \left( 3t + 1 \right) ^{3} \left( 373t^{4} + 365t^{3} + 96t^{2} + 2t + 1 \right) \left( 370882981746756t^{14} + 1235821177778136t^{13} + 1825384823542621t^{12} + 1618925010561173t^{11} + 984437248019802t^{10} + 438863537487496t^{9} + 146908410236867t^{8} + 36698128280865t^{7} + 6666634373419t^{6} + 845208021602t^{5} + 70757230470t^{4} + 3686917584t^{3} + 110512155t^{2} + 439488t + 2574 \right)D^{6} + \left( 3t + 1 \right) ^{2} \left( 1917383421374744233680t^{22} + 9593215233427098537552t^{21} + 22141426943269327633824t^{20} + 31522537204847765091644t^{19} + 31243486067577275895155t^{18} + 23051450223207943854908t^{17} + 13186455334220342501446t^{16} + 5989765156637178019462t^{15} + 2184653853773713311361t^{14} + 640667374859373081855t^{13} + 150178602185485837440t^{12} + 27830075059075270443t^{11} + 4018205515080241591t^{10} + 445011913288498555t^{9} + 37422288339591265t^{8} + 2426057314022834t^{7} + 132643421153357t^{6} + 7344039412274t^{5} + 414271277979t^{4} + 17169487395t^{3} + 442160058t^{2} + 1331334t + 5148 \right)D^{5} + \left( 3t + 1 \right) \left( 47934585534368605842000t^{23} + 244486361981960675320320t^{22} + 577451782367279535842712t^{21} + 845558954923340138870772t^{20} + 867131075892538217769329t^{19} + 666258158716167170054481t^{18} + 399581609435490641238812t^{17} + 191621285205396043502812t^{16} + 74351663585330003918031t^{15} + 23405406690772319041012t^{14} + 5954368322344001667657t^{13} + 1213657885157886256314t^{12} + 195744893173345827789t^{11} + 24581798464810595335t^{10} + 2353583538151343811t^{9} + 166361704297566469t^{8} + 8082211588479482t^{7} + 205844071581759t^{6} - 3219721988566t^{5} - 489416952345t^{4} - 18083607999t^{3} - 375141939t^{2} - 953154t - 2574 \right)D^{4} + t \left( 603975777733044433609200t^{23} + 3169763501301840051920880t^{22} + 7732030070524365974373840t^{21} + 11744698723784710157430972t^{20} + 12554749243337302047900249t^{19} + 10106134955256535325445038t^{18} + 6383056693161929776850843t^{17} + 3241713422593201074132426t^{16} + 1340665224590672655915405t^{15} + 453353806473176294372529t^{14} + 125104224655330161364949t^{13} + 27997872618657681275040t^{12} + 5033770982324695662531t^{11} + 718202560671189838998t^{10} + 80093769416559248842t^{9} + 6849962246411100793t^{8} + 437375413743273552t^{7} + 19888528622349084t^{6} + 584502919125204t^{5} + 9156033307483t^{4} + 59786940567t^{3} - 14541201t^{2} + 7589346t + 7254 \right)D^{3} + 2 t ^{2} \left( 667249430638410993320640t^{22} + 3407091293783698358345208t^{21} + 8072894536919176329769536t^{20} + 11912459394634428621466062t^{19} + 12389514058649865519080466t^{18} + 9724910695790777990330272t^{17} + 6001063350241047741844114t^{16} + 2979942192726914522560042t^{15} + 1203987424310763798852531t^{14} + 396810655858786474390794t^{13} + 106355059935871845003660t^{12} + 23022809232209796861264t^{11} + 3985908689665303011990t^{10} + 545052467825173224747t^{9} + 57960276520393873856t^{8} + 4695634279491985469t^{7} + 280642109192263190t^{6} + 11616676102465116t^{5} + 290050955665188t^{4} + 3336307837641t^{3} + 28584219861t^{2} + 572206761t + 3815064 \right)D^{2} + 4 t ^{2} \left( 362385466639826660165520t^{22} + 1812695665280366734384248t^{21} + 4201341754083939945658368t^{20} + 6064593429051310973029470t^{19} + 6177973263837030918206517t^{18} + 4758159297275745387831776t^{17} + 2884965916404478287958764t^{16} + 1407944040555530059498494t^{15} + 558424351869508647169156t^{14} + 180246995483444319757251t^{13} + 47164221760415505401373t^{12} + 9931539447857793170955t^{11} + 1666203349867055552559t^{10} + 219916289524462256718t^{9} + 22474447097817438868t^{8} + 1739640077348405244t^{7} + 98218498421749098t^{6} + 3731680018574861t^{5} + 78498071889615t^{4} + 523444357113t^{3} + 3552089178t^{2} + 164620782t + 875160 \right)D + 8 t ^{2} \left( 73956217681597277584800t^{22} + 364738915042597450354320t^{21} + 832481502082979438556288t^{20} + 1183304809709366254442388t^{19} + 1188029587398056616910866t^{18} + 902898448113718916063942t^{17} + 540672402156900214802135t^{16} + 260584647815674869586040t^{15} + 101954215183874332744740t^{14} + 32396990094529857934086t^{13} + 8323544115736870068159t^{12} + 1715894449245990544839t^{11} + 280945550251724292153t^{10} + 36070817234196682149t^{9} + 3573049623764848366t^{8} + 266774470377135991t^{7} + 14390286119070367t^{6} + 509922063844281t^{5} + 9263362676016t^{4} + 26676542730t^{3} - 80220330t^{2} + 12974130t + 72072 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ²P2 × Gr(2, 6)

Show multiplicities

Sector dimension: 10
Spectral radius: $22$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$22$	1	1
$0.8799497 - 9.902984\mathrm{i}$	1	1
$0.8799497 + 9.902984\mathrm{i}$	1	1
$-5$	1	1
$-3$	3	1, 2
$-2$	1	1
$-1.87995 - 0.4893211\mathrm{i}$	1	1
$-1.87995 + 0.4893211\mathrm{i}$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right)^{3} \left(\lambda + 5\right) \left(\lambda - 22\right) \left(\lambda^{4} + 2 \lambda^{3} + 96 \lambda^{2} + 365 \lambda + 373\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right)^{2} \left(\lambda + 5\right) \left(\lambda - 22\right) \left(\lambda^{4} + 2 \lambda^{3} + 96 \lambda^{2} + 365 \lambda + 373\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 24 & 108 & 54 & 99 & 150 & 640 & 852 & 2725 \\ 1 & 0 & 5 & 10 & -6 & 22 & 22 & 12 & -3 & 100 \\ 1 & 0 & 2 & 14 & 10 & 16 & 22 & 111 & 162 & 580 \\ 0 & 1 & 1 & 2 & -1 & 5 & 7 & 2 & -6 & -15 \\ 0 & 1 & 0 & 5 & 1 & 2 & 5 & 26 & 32 & 75 \\ 0 & 0 & 1 & 0 & 1 & 0 & 2 & 4 & 10 & 60 \\ 0 & 0 & 1 & 0 & 0 & 0 & -3 & 10 & 14 & 45 \\ 0 & 0 & 0 & \frac{1}{11} & -\frac{10}{11} & \frac{32}{11} & \frac{21}{11} & -\frac{50}{11} & -\frac{48}{11} & -\frac{100}{11} \\ 0 & 0 & 0 & \frac{26}{11} & \frac{15}{11} & -\frac{4}{11} & \frac{18}{11} & \frac{75}{11} & \frac{72}{11} & \frac{260}{11} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{8}{5} & \frac{9}{5} & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta