Family 2-270-60-a

← 2-272-60-h · 2-270-60-b →

A smooth divisor of bidegree $(2,1)$ in $\mathbb P^2\times\mathbb P^3$.

$\pi_1\colon X\to\mathrm{Gr}(2,7)$ is a closed embedding.
$\pi_2\colon X\to\mathbb P^3$ is a conic bundle whose discriminant is a Cayley cubic surface with $4$ ordinary nodes, over which the fibers are double lines.
$\phi\colon X\to\mathbb P^2$ is a $\mathbb P^2$-bundle, the projectivization of the rank-$3$ kernel of a surjection $\mathcal O^{\oplus4}\to\mathcal O(2)$ on $\mathbb P^2$ given by four general conics.

More details (pdf)

ℙ³P3 × Gr(2, 7) · 30601.444wF (preferred model)

Defining data

Ambient: ℙ³P3 × Gr(2, 7)
Bundle: $(E^{ω_{5}})^{\oplus 3} \oplus E^{ω_{3} + ω_{4}}$ 3 copies of E^(w5) + E^(w3 + w4)
Bundle: $O_{2} (1)^{\oplus 3} \oplus Q_{1} ⊠ U_{2}^{\lor}$ 3 copies of O_2(1) + Q_1 box U_2^v
ZL62 label: 30601.444wF

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;7) \\ \det\mathcal{E} &\cong \mathcal{O}(2;6) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	60
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	270
$\chi(X, \mathrm{T}_X)$	=	0

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	270
$\mathrm{c}_1^2\mathrm{c}_2$	=	168
$\mathrm{c}_2^2$	=	91
$\mathrm{c}_1\mathrm{c}_3$	=	54
$\mathrm{c}_4$	=	9

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		3		0		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	0	x₀	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 52	0	0	x₁ - 12	64
$\mathrm{HH}^{ 4 }$ = 60	0	0	0	0	60
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 55t + 158t^{2} + 55t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(65)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(107)

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( 14675t^{8} + 26840t^{7} + 28773t^{6} + 12874t^{5} + 2361t^{4} + 81t^{3} + 8t^{2} + 2t - 1 \right) \left( 24497420682832500t^{13} + 93137273471519420t^{12} + 136177543013277909t^{11} + 110968261166543157t^{10} + 58596242888203113t^{9} + 21417481721823210t^{8} + 5491558126643825t^{7} + 963372809724021t^{6} + 110374727051161t^{5} + 8044514164799t^{4} + 408238480467t^{3} + 14291290467t^{2} - 30089961t + 1115370 \right)D^{6} + \left( 30197970475727622750000t^{22} + 171099701422621069325500t^{21} + 433642012708595379565500t^{20} + 674847873226456280407745t^{19} + 724243512208879556235020t^{18} + 567629947816201118653830t^{17} + 336242227587210615447702t^{16} + 153744838901951013453329t^{15} + 54824589581663273709449t^{14} + 15255546801234256103195t^{13} + 3288987456825596432379t^{12} + 543899387021715706440t^{11} + 69049449428955557005t^{10} + 7101063980353270046t^{9} + 706953587806001715t^{8} + 81465148566407983t^{7} + 9400590650485333t^{6} + 836713700471082t^{5} + 50009529807579t^{4} + 2040045573555t^{3} + 57332417310t^{2} - 94731363t + 2230740 \right)D^{5} + \left( 251649753964396856250000t^{22} + 1382557425326244467072500t^{21} + 3326186411069059626613200t^{20} + 4858897478236415900897065t^{19} + 4888077881065742138334273t^{18} + 3606362603141599967505332t^{17} + 2023042788854564469513395t^{16} + 880695623351726706548785t^{15} + 300029855548194582271759t^{14} + 79840085125059683213460t^{13} + 16430984026893570158001t^{12} + 2573143320127375416302t^{11} + 299525864720357502981t^{10} + 24685591799163020047t^{9} + 1168248320303932985t^{8} - 28569371747956470t^{7} - 12049067870213715t^{6} - 1181331015135029t^{5} - 59561301122006t^{4} - 1893382151013t^{3} - 50324152059t^{2} + 104757543t - 1115370 \right)D^{4} + t \left( 1056928966650466796250000t^{21} + 5663938900824754096592500t^{20} + 13067942183998120418174500t^{19} + 18134056688762014789936875t^{18} + 17290888189385910813030560t^{17} + 12119224207360583796741806t^{16} + 6484052469097213803538010t^{15} + 2701571709800622169100791t^{14} + 882620209454874928799003t^{13} + 225369360766764531646057t^{12} + 44534220870701498865655t^{11} + 6731224794347225908040t^{10} + 772269505184193928245t^{9} + 67320234047907480970t^{8} + 4460994104215359843t^{7} + 214044324693670731t^{6} + 6087540777882615t^{5} + 49223570418894t^{4} - 537554917857t^{3} - 37045774251t^{2} + 437659650t + 780759 \right)D^{3} + 2 t ^{2} \left( 1167654858394801413000000t^{20} + 6136601996700102630248000t^{19} + 13703479158790431758072000t^{18} + 18267276027658038131946260t^{17} + 16690669006399011153534536t^{16} + 11218610283807745610118128t^{15} + 5767420241405369590633640t^{14} + 2312988030479820428405360t^{13} + 727955040811878652541984t^{12} + 179049987852811988275869t^{11} + 34076313825933300908418t^{10} + 4968028962470223793911t^{9} + 552818537962398662110t^{8} + 47150451768357498971t^{7} + 3056094151589527767t^{6} + 138563049535304654t^{5} + 3363958685110472t^{4} + 14723567220731t^{3} + 183443663234t^{2} + 5472865386t + 165282120 \right)D^{2} + 4 t ^{2} \left( 634157379990280077750000t^{20} + 3284770613356408292525500t^{19} + 7158469759677776385540500t^{18} + 9260614812067264780675645t^{17} + 8192588233408801161334460t^{16} + 5330384785452109899361389t^{15} + 2653862036103392419236644t^{14} + 1031039104089979189753862t^{13} + 314313323018787660863937t^{12} + 74864168865638011810587t^{11} + 13800171010135098357336t^{10} + 1952663835909934237544t^{9} + 212039347757253154121t^{8} + 17769758926368157566t^{7} + 1126135322321556324t^{6} + 47926538848320447t^{5} + 944205614696538t^{4} - 2833990934958t^{3} + 27997695288t^{2} + 2668239630t + 40153320 \right)D + 24 t ^{2} \left( 43139957822468032500000t^{20} + 221245955677531289765000t^{19} + 474145079579690255378200t^{18} + 600816519598074441627290t^{17} + 519655683745132856002638t^{16} + 330315705494262793844281t^{15} + 160600507296334790847948t^{14} + 60895173334944696079445t^{13} + 18104551657146915044376t^{12} + 4203339034983738924544t^{11} + 755451903958018140523t^{10} + 104426531844390345937t^{9} + 11131387220058461446t^{8} + 920655394240251492t^{7} + 57312744436626457t^{6} + 2321909171218348t^{5} + 38521003934502t^{4} - 350231422077t^{3} - 942523182t^{2} + 54182196t + 1487160 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ³P3 × Gr(2, 7)

Sector dimension: 9
Spectral radius: $9.18182555$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$9.181826$	1	1
$1.500668 - 6.814779\mathrm{i}$	1	1
$1.500668 + 6.814779\mathrm{i}$	1	1
$-4$	1	1
$-3.993981$	1	1
$-2.536063 - 1.422394\mathrm{i}$	1	1
$-2.536063 + 1.422394\mathrm{i}$	1	1
$-0.5585275 - 0.8124381\mathrm{i}$	1	1
$-0.5585275 + 0.8124381\mathrm{i}$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right) \left(\lambda^{8} - 2 \lambda^{7} - 8 \lambda^{6} - 81 \lambda^{5} - 2361 \lambda^{4} - 12874 \lambda^{3} - 28773 \lambda^{2} - 26840 \lambda - 14675\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 4 & 3 & 3 & 3 & 16 & 36 & 60 \\ 2 & 0 & 3 & 4 & 0 & 6 & -3 & -9 & -12 \\ 1 & 0 & 0 & 0 & 0 & 0 & 3 & 9 & 24 \\ 0 & 1 & 3 & 0 & 0 & 3 & 0 & 0 & -9 \\ 0 & 2 & -1 & 3 & 0 & 3 & 4 & 6 & 0 \\ 0 & 0 & 1 & 0 & 0 & -2 & 0 & 0 & 9 \\ 0 & 0 & 0 & -1 & -5 & 2 & -6 & -9 & -12 \\ 0 & 0 & 0 & \frac{11}{3} & 4 & \frac{8}{3} & 4 & 6 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{3} & 4 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta