Family 3-262-58-a

← 3-266-59-a · 3-261-58-a →

Blow-up of $\mathbb P^2\times\mathbb P^2$ along a smooth rational curve.

$\pi_1\colon X\to F$ is the blow-up of the Fano fourfold 2-432-90-d, a divisor of bidegree $(1,1)$ in $\mathbb P^2\times\mathbb P^3$, along a smooth quadric surface $\mathbb P^1\times\mathbb P^1$.
$\pi_2\colon X\to G$ is the blow-up of another copy of 2-432-90-d, a divisor of bidegree $(1,1)$ in $\mathbb P^2\times\mathbb P^3$, along a smooth quadric surface $\mathbb P^1\times\mathbb P^1$.
$\pi_3\colon X\to\mathbb P^2\times\mathbb P^2$ is the blow-up of $\mathbb P^2\times\mathbb P^2$ (2-486-100-a) along a smooth rational curve of bidegree $(2,2)$, the graph of an isomorphism between a conic in the first factor and a conic in the second.

More details (pdf)

Varieties in this family can be blown down to

ℙ²P2 × Fl(1, 3, 4) · 2034.2b (preferred model)

Defining data

Ambient: ℙ²P2 × Fl(1, 3, 4)
Bundle: $E^{ω_{3}} \oplus E^{ω_{1} + ω_{4}}$ E^(w3) + E^(w1 + w4)
Bundle: $O_{2} (1, 0) \oplus O_{1} (1) ⊠ U_{2, 2}^{\lor} \otimes U_{2, 3}^{\lor}$ O_2(1,0) + O_1(1) box U_2,2^v ot U_2,3^v
ZL62 label: 2034.2b

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3,3) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2,1) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	58
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	262
$\chi(X, \mathrm{T}_X)$	=	3

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	262
$\mathrm{c}_1^2\mathrm{c}_2$	=	160
$\mathrm{c}_2^2$	=	99
$\mathrm{c}_1\mathrm{c}_3$	=	58
$\mathrm{c}_4$	=	13

Hodge diamond

				1
			0		0
		0		3		0
	0		0		0		0
0		0		5		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 + x₁ - 3	0	x₀ - 3	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 43	0	0	x₁ - 16	59
$\mathrm{HH}^{ 4 }$ = 58	0	0	0	0	58
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 53t + 154t^{2} + 53t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	No match
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( 5t + 1 \right) \left( 7t^{2} + t + 1 \right) \left( 8t^{2} + 4t + 1 \right) \left( 832t^{5} + 512t^{4} + 296t^{3} - 28t^{2} + 8t - 1 \right) \left( 756444107489280t^{17} + 2435045902901248t^{16} + 3300467735558144t^{15} + 1541732517094400t^{14} - 1230425939540480t^{13} - 2229035916580096t^{12} - 1482631891594752t^{11} - 483198222660352t^{10} - 76270218578720t^{9} + 3365012365008t^{8} + 5421061380416t^{7} + 1655000018416t^{6} + 251921468176t^{5} + 24286859480t^{4} + 1376412184t^{3} + 56592012t^{2} + 190954t + 5633 \right)D^{6} + 2 \left( 1850322802447378022400t^{27} + 8424222924407507517440t^{26} + 17824544044391319732224t^{25} + 20855593883217324933120t^{24} + 11535764124711081410560t^{23} - 3791732357467886157824t^{22} - 13938507865284456079360t^{21} - 14488440496674619871232t^{20} - 9533204885973328918528t^{19} - 4446957765627592509440t^{18} - 1504435081787456610304t^{17} - 348135357783255918848t^{16} - 37872154237361740672t^{15} + 9090578944569215808t^{14} + 6210329809627184000t^{13} + 1913420038910728048t^{12} + 404214051721531216t^{11} + 63506575291576544t^{10} + 7665939838500528t^{9} + 775788337270280t^{8} + 80196331110816t^{7} + 9779606153776t^{6} + 1005110767064t^{5} + 75466458669t^{4} + 3451524645t^{3} + 113128262t^{2} + 275165t + 5633 \right)D^{5} + \left( 30838713374122967040000t^{27} + 137332886946567002521600t^{26} + 279763331812761969098752t^{25} + 300403078901367017308160t^{24} + 118321826501462196813824t^{23} - 129846404573763192815616t^{22} - 259341710874921824256000t^{21} - 229079282344978458558464t^{20} - 132081683693796158189568t^{19} - 53248004532302529814528t^{18} - 14755421950438534574080t^{17} - 2266275628300971849728t^{16} + 195637225251988698112t^{15} + 257996110646559140096t^{14} + 96195349947680996608t^{13} + 24050943176653867904t^{12} + 4501415818747819280t^{11} + 646867665433634992t^{10} + 70803464797385856t^{9} + 5630705985808008t^{8} + 249006494578072t^{7} - 8564369422428t^{6} - 2334802990000t^{5} - 185081460214t^{4} - 6639023267t^{3} - 202153272t^{2} - 305884t - 5633 \right)D^{4} + 2 t \left( 64761298085658230784000t^{26} + 283332195628419802726400t^{25} + 560932311913683877888000t^{24} + 560426364396015486500864t^{23} + 140868121298769503256576t^{22} - 361743491036272948084736t^{21} - 568151582754689513381888t^{20} - 447374906057719888211968t^{19} - 230869524379856286906368t^{18} - 81292049598286621999104t^{17} - 18017036156654044666880t^{16} - 1011078818062632041216t^{15} + 1090280684566499937152t^{14} + 570847846592353621696t^{13} + 172255927216053863872t^{12} + 37700264684640242416t^{11} + 6289528100136106528t^{10} + 813205179714520496t^{9} + 81878675845554288t^{8} + 6380036235367680t^{7} + 370533364772872t^{6} + 15215708823476t^{5} + 335800374260t^{4} + 4037040163t^{3} + 42228466t^{2} + 1372866t + 1419 \right)D^{3} + 4 t ^{2} \left( 71545815027965283532800t^{25} + 308731907145262185840640t^{24} + 598474857142715494301696t^{23} + 564132870627956177108992t^{22} + 72603950972068689788928t^{21} - 463820823582430503927808t^{20} - 636832924967094188142592t^{19} - 459566962931512147726336t^{18} - 215647955323104226950144t^{17} - 66843809383874576635392t^{16} - 11281597321390671662080t^{15} + 1027104243633414129280t^{14} + 1495004008144820699008t^{13} + 607383013080084971968t^{12} + 163750153121014415728t^{11} + 32947457617017975816t^{10} + 5100070059498806752t^{9} + 615711129414195372t^{8} + 58194791911139844t^{7} + 4284018981554370t^{6} + 231383549466060t^{5} + 8535270615031t^{4} + 145671437174t^{3} + 2035660668t^{2} + 48655798t + 560418 \right)D^{2} + 16 t ^{2} \left( 19428389425697469235200t^{25} + 82984427511503422750720t^{24} + 158464383189055680479232t^{23} + 142859732305982761730048t^{22} + 4199414978712711798784t^{21} - 137330023580093422305280t^{20} - 173445667510598038038528t^{19} - 117188128727255565267968t^{18} - 50803511007105534393344t^{17} - 13995487119737696337664t^{16} - 1643455003260746515456t^{15} + 591784976284985659840t^{14} + 437053295827342688832t^{13} + 156997994159040898304t^{12} + 39452345988796548304t^{11} + 7482025612679886048t^{10} + 1095694727275346248t^{9} + 125600002987209488t^{8} + 11263386425209204t^{7} + 787237759480116t^{6} + 39549933445862t^{5} + 1308050315362t^{4} + 16430197032t^{3} + 259765977t^{2} + 6995136t + 61963 \right)D + 16 t ^{2} \left( 7929954867631620096000t^{25} + 33636182520494397849600t^{24} + 63590528011361687175168t^{23} + 55563619573848654348288t^{22} - 2403827139920602497024t^{21} - 58967783505780630257664t^{20} - 70729231549785455575040t^{19} - 45627370992908612929536t^{18} - 18620420914360218133504t^{17} - 4637166029216519847936t^{16} - 325962997620400157696t^{15} + 311220662238447019520t^{14} + 183478795049630102656t^{13} + 61573796401976670720t^{12} + 14796711071487392624t^{11} + 2694657726387379752t^{10} + 379402466987884208t^{9} + 41897111724297436t^{8} + 3605241434216100t^{7} + 241091301420650t^{6} + 11353383253492t^{5} + 342254955311t^{4} + 3321253122t^{3} + 62102974t^{2} + 1394189t + 11266 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ²P2 × Fl(1, 3, 4)

Show multiplicities

Sector dimension: 13
Spectral radius: $9.21615874$
Diagonalizable at 1: Yes
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$9.216159$	1	1
$0.2739864 - 6.140911\mathrm{i}$	1	1
$0.2739864 + 6.140911\mathrm{i}$	1	1
$-5$	2	1, 1
$-2 - 2\mathrm{i}$	1	1
$-2 + 2\mathrm{i}$	1	1
$-0.5 - 2.598076\mathrm{i}$	2	1, 1
$-0.5 + 2.598076\mathrm{i}$	2	1, 1
$-0.8820658 - 1.269297\mathrm{i}$	1	1
$-0.8820658 + 1.269297\mathrm{i}$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 5\right)^{2} \left(\lambda^{2} + \lambda + 7\right)^{2} \left(\lambda^{2} + 4 \lambda + 8\right) \left(\lambda^{5} - 8 \lambda^{4} + 28 \lambda^{3} - 296 \lambda^{2} - 512 \lambda - 832\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 5\right) \left(\lambda^{2} + \lambda + 7\right) \left(\lambda^{2} + 4 \lambda + 8\right) \left(\lambda^{5} - 8 \lambda^{4} + 28 \lambda^{3} - 296 \lambda^{2} - 512 \lambda - 832\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 2 & 0 & 0 & 3 & 3 & 3 & 6 & 16 & 8 & 16 & 10 \\ 1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 3 & 3 & 4 \\ 1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 0 & 6 & 4 \\ 2 & 1 & 1 & 0 & 4 & 2 & 2 & 2 & -2 & 0 & 0 & 0 & 4 \\ 0 & 1 & 1 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 2 & -1 & 1 & 0 & -1 & -1 & 1 & 0 & 0 & 0 & -2 & -3 \\ 0 & 1 & 1 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 6 \\ 0 & 0 & 3 & 1 & 2 & 1 & 1 & -1 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 2 & 0 & 1 & 1 & 1 & 0 & 4 & 2 & 2 & 0 \\ 0 & 0 & 0 & 0 & 3 & 2 & 3 & -1 & -3 & -2 & 0 & -2 & -2 \\ 0 & 0 & 0 & 0 & 0 & -1 & -3 & 5 & 4 & 2 & -1 & 3 & 2 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 4 & 2 & 1 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 3 & 5 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta