Family 2-225-51-a

← 2-240-54-g · 2-225-51-b →

A small resolution of a fourfold obtained from a bisymplectic Grassmannian.

$\pi_1\colon X\to Y$ maps birationally onto a fourfold of degree $33$; it contracts a single plane with normal bundle $\mathcal O(-1)^{\oplus 2}$ to the unique singular point of the image, locally the cone over $\mathbb P^2\times\mathbb P^1$.
$\pi_2\colon X\to Q^4$ maps birationally onto the smooth quadric fourfold (1-512-105-a), with center a non-normal surface of degree $5$, the image of a rational ruled surface whose rulings map to lines, with two points identified to an ordinary double point; the fibers are lines over the rest of the surface and $\mathbb P^1\times\mathbb P^1$ over the double point.

More details (pdf)

Varieties in this family can be blown down to

1-512-105-a

Fl(2, 4, 7) · 609.44000 (preferred model)

Defining data

Ambient: Fl(2, 4, 7)
Bundle: $(E^{ω_{1}})^{\oplus 3} \oplus (E^{ω_{5}})^{\oplus 2}$ 3 copies of E^(w1) + 2 copies of E^(w5)
Bundle: $(U_{3}^{\lor})^{\oplus 3} \oplus (Q_{1}^{\lor} \otimes ⋀^{2} U_{2}^{\lor} \otimes ⋀^{2} U_{3}^{\lor})^{\oplus 2}$ 3 copies of U_3^v + 2 copies of Q_1^v ot wedge^2 U_2^v ot wedge^2 U_3^v
ZL62 label: 609.44000

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4,5) \\ \det\mathcal{E} &\cong \mathcal{O}(3,4) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	51
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	225
$\chi(X, \mathrm{T}_X)$	=	-2

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	225
$\mathrm{c}_1^2\mathrm{c}_2$	=	150
$\mathrm{c}_2^2$	=	102
$\mathrm{c}_1\mathrm{c}_3$	=	50
$\mathrm{c}_4$	=	11

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		5		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 + 2	0	x₀ + 2	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 43	0	0	x₁ - 5	48
$\mathrm{HH}^{ 4 }$ = 51	0	0	0	0	51
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 46t + 131t^{2} + 46t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

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( 5033t^{11} + 1137t^{10} - 2814t^{9} + 9038t^{8} - 1525t^{7} - 2124t^{6} + 4664t^{5} + 3821t^{4} + 856t^{3} + 39t^{2} - 5t - 1 \right) \left( 381286565580276t^{21} + 1171040825665504t^{20} + 485895194223504t^{19} - 4967207611953532t^{18} - 9990027381034246t^{17} - 8964542255764404t^{16} - 4289508301195686t^{15} - 1759380669282774t^{14} - 1750043604353965t^{13} - 2355138935485458t^{12} - 2447385330144835t^{11} - 1521175061350935t^{10} - 418293606408290t^{9} + 22496860985048t^{8} + 44821650341924t^{7} + 9433735993304t^{6} - 182764740919t^{5} - 330888694374t^{4} - 32974599321t^{3} - 2845609443t^{2} - 21082356t - 437400 \right)D^{6} + \left( 40299320975876111268t^{32} + 137034554488739750708t^{31} + 66265038184131147768t^{30} - 604621860532768065624t^{29} - 1287527772701532777450t^{28} - 1151188546561996792758t^{27} - 883461645702951962310t^{26} - 1043428107693975676450t^{25} - 1061788873719450437623t^{24} - 664584943255785754945t^{23} - 391295022591366732293t^{22} - 282461149842005912201t^{21} - 207315641514580200147t^{20} - 210234818729985482820t^{19} - 150674978768501420305t^{18} - 49331475664628189901t^{17} - 42304448430379166747t^{16} - 61044990907612876993t^{15} - 38212002911728161392t^{14} - 8340007115143190845t^{13} + 2161690445239184874t^{12} + 1731533171431628257t^{11} + 370830854210487545t^{10} + 165327330312079t^{9} - 14265316545937300t^{8} - 2711851029075668t^{7} - 282792723655721t^{6} - 29016946190782t^{5} - 2779936982181t^{4} - 172728200025t^{3} - 11379377592t^{2} - 58873068t - 874800 \right)D^{5} + \left( 335827674798967593900t^{32} + 1169686335241600498460t^{31} + 602317050115865820936t^{30} - 5622465243337635435692t^{29} - 12642403725372720210842t^{28} - 12017349130342273792630t^{27} - 7211635687669402590826t^{26} - 6041473929761257388012t^{25} - 7050978138022265005419t^{24} - 6344125054373999875497t^{23} - 5146336480190963033941t^{22} - 3466712577932804677698t^{21} - 1880155259206688007190t^{20} - 1647004890779634175372t^{19} - 1259537361390463855057t^{18} - 256936783475201894474t^{17} + 171190162901825487421t^{16} + 81278128175780656663t^{15} + 5112514240705847379t^{14} + 10086038296753955346t^{13} + 13209225316568622266t^{12} + 5721657768408385638t^{11} + 922403885926565068t^{10} - 88848123939911890t^{9} - 61611837312343300t^{8} - 11367042594560014t^{7} - 1284259428307764t^{6} - 74864788674736t^{5} + 727386639642t^{4} + 129664394784t^{3} + 11111784921t^{2} + 67288968t + 437400 \right)D^{4} + t \left( 1410476234155663894380t^{31} + 5004197265871660593980t^{30} + 2662358256708120560040t^{29} - 25525624552354164702264t^{28} - 59338080828874686649998t^{27} - 59285759995619109912378t^{26} - 33386146981586555103906t^{25} - 21251581885763276883486t^{24} - 25863837738693847210669t^{23} - 30437491211464795214883t^{22} - 31151928746842395656991t^{21} - 22565948686147806185747t^{20} - 10363052773640195772717t^{19} - 6021380784035277009952t^{18} - 4668830014512615520735t^{17} - 1335590813066712162527t^{16} + 697055144370606377831t^{15} + 656793843943108503095t^{14} + 212787817568193902368t^{13} + 51229053044732904177t^{12} + 21968203290171968242t^{11} + 8261823484503367193t^{10} + 1026881042589330339t^{9} - 266263712685090895t^{8} - 106582475650998148t^{7} - 16973065416063938t^{6} - 1926923585649787t^{5} - 110443160418170t^{4} - 2172064846743t^{3} - 47061665505t^{2} - 530672436t - 192456 \right)D^{3} + 2 t ^{2} \left( 1558240411067209635696t^{30} + 5605798178342056388728t^{29} + 3033079809829206177072t^{28} - 29847383375501465574184t^{27} - 70871427598769185022968t^{26} - 73410255693984176957000t^{25} - 41894457137630990943100t^{24} - 24092952521162170063710t^{23} - 28568454683617596223206t^{22} - 37809742589864033770060t^{21} - 42738506155769986579033t^{20} - 31910766508565697395653t^{19} - 12897147558074529970398t^{18} - 4600834433393173627520t^{17} - 3703784721484612577920t^{16} - 1597925214874999958942t^{15} + 342763766499527074869t^{14} + 570255431388543949641t^{13} + 194605758254648964132t^{12} + 26803300981896848748t^{11} + 5652799424322104598t^{10} + 3293906723233743083t^{9} + 501287305809547820t^{8} - 121451216711568788t^{7} - 47339727005634651t^{6} - 8134550810411485t^{5} - 1015867964669385t^{4} - 51781226118603t^{3} - 1022771168496t^{2} - 26334829107t - 206643150 \right)D^{2} + 4 t ^{2} \left( 846285740493398336628t^{30} + 3075314110091015618108t^{29} + 1678216343200388131848t^{28} - 16882330900186292098920t^{27} - 40641029404114966013970t^{26} - 43149526784443794644454t^{25} - 25419018267116244352330t^{24} - 14653832701248123544856t^{23} - 16757265266405438897209t^{22} - 22789957341033273031257t^{21} - 26819489800044036950676t^{20} - 20375603858435457685350t^{19} - 7601148096167604866867t^{18} - 1451849461933391569176t^{17} - 1196741133939497616927t^{16} - 833415239364962339799t^{15} - 10971193949764049450t^{14} + 204595267871805960770t^{13} + 72343343769824426458t^{12} + 3163208156643565535t^{11} - 1335344333688344408t^{10} + 665990086341946024t^{9} + 198320201029383311t^{8} - 14399411930692187t^{7} - 9437082518093099t^{6} - 2137877456283767t^{5} - 297631366328466t^{4} - 13457004434055t^{3} - 294076951263t^{2} - 7432531812t - 41990400 \right)D + 16 t ^{2} \left( 86355687805448809860t^{30} + 315929887343631892860t^{29} + 173144247148985023944t^{28} - 1769877580332043523952t^{27} - 4297323429259672916534t^{26} - 4634780719525072266354t^{25} - 2805861913810272263178t^{24} - 1663296356053963662812t^{23} - 1856828640358573090687t^{22} - 2515380016268342366765t^{21} - 3006742121756384159854t^{20} - 2308994484590926766744t^{19} - 824664893287780635991t^{18} - 69898914067054708048t^{17} - 56297811804078110989t^{16} - 77791368171598215465t^{15} - 14152445269542239637t^{14} + 13131463482389162639t^{13} + 5000293839689471638t^{12} - 291811166929643749t^{11} - 321025672818719659t^{10} + 25856791753530302t^{9} + 18701914359818236t^{8} + 1095940568167014t^{7} - 244726187287563t^{6} - 119319184346706t^{5} - 18607143232404t^{4} - 757423118754t^{3} - 19272729249t^{2} - 435816612t - 1749600 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 4, 7)

Sector dimension: 11
Spectral radius: $10.4104298$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$10.41043$	1	1
$-4.808096 - 5.878882\mathrm{i}$	1	1
$-4.808096 + 5.878882\mathrm{i}$	1	1
$-3.295174$	1	1
$-2.00898 - 0.8553535\mathrm{i}$	1	1
$-2.00898 + 0.8553535\mathrm{i}$	1	1
$0.8282453 - 0.5444115\mathrm{i}$	1	1
$0.8282453 + 0.5444115\mathrm{i}$	1	1
$0.2821727 - 0.8331239\mathrm{i}$	1	1
$0.2821727 + 0.8331239\mathrm{i}$	1	1
$-0.7019407$	1	1

Characteristic polynomial

$$\begin{equation}\lambda^{11} + 5 \lambda^{10} - 39 \lambda^{9} - 856 \lambda^{8} - 3821 \lambda^{7} - 4664 \lambda^{6} + 2124 \lambda^{5} + 1525 \lambda^{4} - 9038 \lambda^{3} + 2814 \lambda^{2} - 1137 \lambda - 5033\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 9 & 30 & 6 & 27 & 21 & 12 & 4 & 5 \\ 1 & 0 & 3 & -2 & -6 & 0 & -6 & -4 & 12 & 6 & 4 \\ 1 & 0 & -1 & 4 & 10 & 2 & 8 & 6 & 9 & 3 & 0 \\ 0 & 1 & 0 & 1 & 3 & 0 & 3 & 2 & 0 & 0 & 0 \\ 0 & 1 & 1 & -1 & -1 & 0 & 2 & 2 & 2 & 2 & 3 \\ 0 & 1 & 0 & 0 & 3 & 0 & 0 & 0 & -10 & -6 & -3 \\ 0 & 0 & 1 & 0 & 0 & 0 & -3 & -2 & -2 & -2 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 8 & 4 & 0 \\ 0 & 0 & 0 & 1 & 11 & \frac{3}{2} & \frac{27}{2} & \frac{23}{2} & -1 & 0 & 4 \\ 0 & 0 & 0 & 1 & -14 & -\frac{1}{2} & -\frac{43}{2} & -\frac{37}{2} & 3 & 0 & -8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 4 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta