Family 2-177-42-b

← 2-177-42-a · 2-176-42-a →

Blow-up of $\mathbb P^4$ along a non-normal sextic surface with two improper double points.

$\pi_1\colon X\to\mathbb P^4$ maps birationally onto $\mathbb P^4$ (1-625-126-a) with center a non-normal sextic surface with two improper double points; its normalization is the blow-up of $\mathbb P^2$ in $7$ points, a conic bundle over $\mathbb P^1$ with six singular fibers. The fibers are smooth quadric surfaces over the two double points and lines over the rest of the surface.
$\pi_2\colon X\to Y\subset\mathrm{Gr}(2,8)$ is birational onto its image and an isomorphism off a plane with normal bundle $\mathcal O(-1)^{\oplus 2}$, which it contracts to a point.

More details (pdf)

Varieties in this family can be blown down to

1-625-126-a

Fl(2, 4, 8) · 709.66611w2 (preferred model)

Defining data

Ambient: Fl(2, 4, 8)
Bundle: $(E^{ω_{2}})^{\oplus 2} \oplus (E^{ω_{7}})^{\oplus 3} \oplus E^{ω_{1} + ω_{4}}$ 2 copies of E^(w2) + 3 copies of E^(w7) + E^(w1 + w4)
Bundle: $O (1, 0)^{\oplus 2} \oplus (⋀^{3} Q_{1}^{\lor} \otimes ⋀^{2} U_{2}^{\lor} \otimes ⋀^{2} U_{3}^{\lor})^{\oplus 3} \oplus ⋀^{2} U_{2}^{\lor} \otimes Σ_{(2, 1)} U_{3}^{\lor}$ 2 copies of O(1,0) + 3 copies of wedge^3 Q_1^v ot wedge^2 U_2^v ot wedge^2 U_3^v + wedge^2 U_2^v ot Sigma_(2,1) U_3^v
ZL62 label: 709.66611w2

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4,6) \\ \det\mathcal{E} &\cong \mathcal{O}(3,5) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	42
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	177
$\chi(X, \mathrm{T}_X)$	=	-11

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	177
$\mathrm{c}_1^2\mathrm{c}_2$	=	138
$\mathrm{c}_2^2$	=	108
$\mathrm{c}_1\mathrm{c}_3$	=	38
$\mathrm{c}_4$	=	17

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		11		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 + 11	0	x₀ + 11	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 37	0	0	x₁ + 4	33
$\mathrm{HH}^{ 4 }$ = 42	0	0	0	0	42
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 37t + 101t^{2} + 37t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(253)
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( 3t + 1 \right) \left( t + 1 \right) ^{2} \left( 1087t^{6} + 459t^{5} + 1091t^{4} + 1005t^{3} + 41t^{2} + 2t - 1 \right) \left( 169958568141376t^{17} + 877620841827744t^{16} + 2285290727910548t^{15} + 4801767147612412t^{14} + 8555274492925296t^{13} + 11255245579709088t^{12} + 10097781984229822t^{11} + 6019432896917663t^{10} + 2355226860011426t^{9} + 613016243223861t^{8} + 125440011443374t^{7} + 30130486861432t^{6} + 7698931785370t^{5} + 1329056097464t^{4} + 131124577336t^{3} + 7329937589t^{2} + 45753820t + 792075 \right)D^{6} + 2 \left( t + 1 \right) \left( 11638932704889569856t^{26} + 81997549086956051360t^{25} + 293382920576418269084t^{24} + 775654462993676809794t^{23} + 1696582326209666196046t^{22} + 3036741689887817512336t^{21} + 4362330113259340856998t^{20} + 5054012484755740219003t^{19} + 4784045567270227803225t^{18} + 3712760858084142404785t^{17} + 2337176219683527637452t^{16} + 1172124123116259225874t^{15} + 462454718180640775512t^{14} + 145328215183899607105t^{13} + 38974142778849629300t^{12} + 10083845990616013740t^{11} + 2611298196536565475t^{10} + 595671178163476419t^{9} + 104074046229484767t^{8} + 13000564771716828t^{7} + 1151803691517516t^{6} + 77394170969587t^{5} + 4867872181764t^{4} + 322770278217t^{3} + 14571908898t^{2} + 63878280t + 792075 \right)D^{5} + \left( 193982211748159497600t^{27} + 1502534176670875019840t^{26} + 5812171139289865828232t^{25} + 16186997124185429975724t^{24} + 36981133537261505833540t^{23} + 69526104615115823285436t^{22} + 104670725628476726720000t^{21} + 125008866580835197413160t^{20} + 119269902643285205674897t^{19} + 91913718157915633829082t^{18} + 57574225361541787336678t^{17} + 29283307163715073563830t^{16} + 12056636810065353241613t^{15} + 4065129164995026893307t^{14} + 1187123295598765142321t^{13} + 334184044902357995865t^{12} + 95348826209449487491t^{11} + 24934006649221140427t^{10} + 5219033168490968265t^{9} + 808144015144001733t^{8} + 88652649246062251t^{7} + 6360720490842469t^{6} + 213040958592515t^{5} - 4211067187677t^{4} - 564662751824t^{3} - 28325089217t^{2} - 137228715t - 792075 \right)D^{4} + 2 t \left( 407362644671134944960t^{26} + 3059499913240026740960t^{25} + 11506878302288578716100t^{24} + 31613597564157668003602t^{23} + 71925462808027512666552t^{22} + 133671066182505802420994t^{21} + 195366901664694449011614t^{20} + 221598679012287179471051t^{19} + 196179117081008763139078t^{18} + 137205559338390226521464t^{17} + 76677124552111972683740t^{16} + 34547284643381022095662t^{15} + 12721734726002040994795t^{14} + 3993854280535002507435t^{13} + 1181781951530493420235t^{12} + 364444735266701888217t^{11} + 111348933325858286130t^{10} + 28957101998310479424t^{9} + 5761296384810618222t^{8} + 837093240395436106t^{7} + 86634711419587831t^{6} + 5990613066756729t^{5} + 225641692535273t^{4} + 3455653367922t^{3} + 60771496526t^{2} + 558133410t + 286400 \right)D^{3} + 4 t ^{2} \left( 450038731255730034432t^{25} + 3299026667227248984656t^{24} + 12143012573553729568196t^{23} + 33091302483130039218330t^{22} + 75241853531298285606624t^{21} + 138682690889355851576220t^{20} + 197983437880788945120378t^{19} + 215629185978420059157051t^{18} + 180011222917822792978889t^{17} + 116543873589799298151247t^{16} + 59320208458236272089094t^{15} + 24154545911247372845606t^{14} + 8143030840751237030626t^{13} + 2465636190560220692972t^{12} + 763157066386786083001t^{11} + 251407258703626889003t^{10} + 77395280104719136616t^{9} + 19269229783676690680t^{8} + 3602286325305997748t^{7} + 491953272565119518t^{6} + 48099391669835022t^{5} + 3103850960159268t^{4} + 105663473083703t^{3} + 1744874662708t^{2} + 35338568727t + 212641605 \right)D^{2} + 8 t ^{2} \left( 244417586802680966976t^{25} + 1759478015627916544016t^{24} + 6374162968631065161784t^{23} + 17287888616721197582316t^{22} + 39336263079165127711316t^{21} + 72058886559142779303242t^{20} + 101030838578293692137704t^{19} + 106698843090233129104658t^{18} + 85216648371028985081308t^{17} + 52027921101411807557181t^{16} + 24639616320459219520712t^{15} + 9279999550223893752532t^{14} + 2943129384146252710306t^{13} + 886637521247183285930t^{12} + 288630075848879249595t^{11} + 97869311248552604247t^{10} + 29385698076948907211t^{9} + 6938441852253355713t^{8} + 1223147340617577906t^{7} + 158372495303939018t^{6} + 14751431913034424t^{5} + 889314669409122t^{4} + 27486164068171t^{3} + 506264575085t^{2} + 9710823115t + 41187900 \right)D + 64 t ^{2} \left( 12470285040953110560t^{25} + 88658179626554058360t^{24} + 317726269845480267072t^{23} + 859386313421564695200t^{22} + 1957222772562960786270t^{21} + 3569988494726219705067t^{20} + 4941879973577232146142t^{19} + 5107341426317284261482t^{18} + 3953752174860870553904t^{17} + 2315142481933037229561t^{16} + 1040822064991953229347t^{15} + 370742519028222604887t^{14} + 113258726679266468431t^{13} + 34531954110775388333t^{12} + 11686204021373842125t^{11} + 3975978673469053297t^{10} + 1153102596862009950t^{9} + 259171450517642082t^{8} + 43462806287497785t^{7} + 5388740056790437t^{6} + 482460484668241t^{5} + 27334975071266t^{4} + 778066226865t^{3} + 16147823601t^{2} + 269591900t + 792075 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 4, 8)

Show multiplicities

Sector dimension: 10
Spectral radius: $12.4153992$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$12.4154$	1	1
$-4.651319 - 7.476138\mathrm{i}$	1	1
$-4.651319 + 7.476138\mathrm{i}$	1	1
$-3$	1	1
$-2$	1	1
$-1.387677$	1	1
$-1$	2	2
$0.137458 - 0.8915824\mathrm{i}$	1	1
$0.137458 + 0.8915824\mathrm{i}$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda + 2\right) \left(\lambda + 3\right) \left(\lambda^{6} - 2 \lambda^{5} - 41 \lambda^{4} - 1005 \lambda^{3} - 1091 \lambda^{2} - 459 \lambda - 1087\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 24 & 48 & 54 & 24 & 56 & 80 & 60 \\ 1 & -1 & 0 & 4 & 12 & 12 & 8 & 30 & 42 & 16 \\ 1 & 4 & 0 & -4 & -8 & -8 & -4 & 6 & 6 & 32 \\ 0 & 1 & 0 & -1 & 0 & 0 & 1 & 4 & 4 & 12 \\ 0 & 1 & 1 & 2 & -1 & 2 & -1 & -8 & -12 & 12 \\ 0 & 1 & 0 & -1 & 0 & -2 & 0 & 8 & 12 & 0 \\ 0 & 0 & 1 & 2 & 4 & 4 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 14 & 12 & 11 & 15 & 24 & -8 \\ 0 & 0 & 0 & 1 & -7 & -5 & -\frac{13}{2} & -10 & -16 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & \frac{7}{2} & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta