Family 2-136-34-b

← 2-136-34-a · 2-132-34-a →

Blow-up of a smooth quadric fourfold along a non-normal surface of degree $8$.

$\varpi_1\colon X\to Y$ is the blow-up of a Gushel–Mukai fourfold $Y$ (a special member of 1-160-39-a) singular exactly along a smooth conic, along that conic; the fibers over the conic are quadric surfaces.
$\varpi_4\colon X\to Q^4$ maps birationally onto the smooth quadric fourfold (1-512-105-a) with center a surface of degree $8$ and class $4h^2$ ($h$ the hyperplane class), smooth except at two non-normal points; the fibers are lines over the smooth points and quadric surfaces over the two special points.

More details (pdf)

Varieties in this family can be blown down to

1-512-105-a
A Gushel–Mukai fourfold singular along a conic

Fl(1, 4, 6) · 58.4325 (preferred model)

Defining data

Ambient: Fl(1, 4, 6)
Bundle: $E^{ω_{3}} \oplus E^{ω_{4}} \oplus E^{ω_{5}} \oplus E^{2 ω_{1}}$ E^(w3) + E^(w4) + E^(w5) + E^(2w1)
Bundle: $⋀^{2} U_{2}^{\lor} \otimes U_{3}^{\lor} \oplus O (0, 1) \oplus Q_{1}^{\lor} \otimes ⋀^{3} U_{2}^{\lor} \otimes U_{3}^{\lor} \oplus O (2, 0)$ wedge^2 U_2^v ot U_3^v + O(0,1) + Q_1^v ot wedge^3 U_2^v ot U_3^v + O(2,0)
ZL62 label: 58.4325

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
$\mathrm{h}^0(X, \omega_X^\vee)$	=	34
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	136
$\chi(X, \mathrm{T}_X)$	=	-15

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	136
$\mathrm{c}_1^2\mathrm{c}_2$	=	124
$\mathrm{c}_2^2$	=	116
$\mathrm{c}_1\mathrm{c}_3$	=	32
$\mathrm{c}_4$	=	20

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		14		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 + x₂ + 15	0	x₀ + 15	x₂
$\mathrm{HH}^{ 3 }$ = x₁ + 19	0	0	x₁	19
$\mathrm{HH}^{ 4 }$ = x₁ - x₂ + 23	0	0	x₁ - x₂ - 11	0	34
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 29t + 76t^{2} + 29t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

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( 2t + 1 \right) \left( 5t + 1 \right) \left( t + 1 \right) ^{2} \left( 4096t^{5} + 4096t^{4} + 1920t^{3} + 192t^{2} - 5t - 1 \right) \left( 28639211028480t^{16} + 108849067458560t^{15} + 192941535264768t^{14} + 203698647171072t^{13} + 139328489467904t^{12} + 69527138484224t^{11} + 31537647569664t^{10} + 14511591365760t^{9} + 5739716329936t^{8} + 1643661697744t^{7} + 315240768569t^{6} + 38443078559t^{5} + 2651439682t^{4} + 54558026t^{3} - 4406791t^{2} - 125281t + 2516 \right)D^{6} + \left( t + 1 \right) \left( 19942055423351193600t^{24} + 116734471259742535680t^{23} + 324216624529549557760t^{22} + 561642542580870676480t^{21} + 673689211760373923840t^{20} + 594437124407823433728t^{19} + 408147919715300802560t^{18} + 234307717753423265792t^{17} + 121651346192477462528t^{16} + 59081507725824339968t^{15} + 25785762035455972864t^{14} + 9448391888383672576t^{13} + 2762269131109554784t^{12} + 625328280589696336t^{11} + 107469969952502042t^{10} + 13767401274047903t^{9} + 1288911154269037t^{8} + 87619946455471t^{7} + 4743860907829t^{6} + 273745830885t^{5} + 15518745259t^{4} + 303634677t^{3} - 19955299t^{2} - 543896t + 7548 \right)D^{5} + \left( 134902139628552192000t^{25} + 858199293633568440320t^{24} + 2576789308431432417280t^{23} + 4814922366144663781376t^{22} + 6223908474911782338560t^{21} + 5906610016496272801792t^{20} + 4342618093657580896256t^{19} + 2653350188446182998016t^{18} + 1463070165756909199360t^{17} + 760283338150800699392t^{16} + 359707760185334989312t^{15} + 144568451679941160448t^{14} + 46803367958082418912t^{13} + 11836777685588716592t^{12} + 2289283178682614366t^{11} + 329676797260968389t^{10} + 33440003948144501t^{9} + 2016514048649894t^{8} + 8631728881146t^{7} - 10658486241220t^{6} - 977830915436t^{5} - 42349414502t^{4} - 830296144t^{3} + 13953879t^{2} + 411695t - 5032 \right)D^{4} + t \left( 463359523071983616000t^{24} + 2777594331446378496000t^{23} + 7883663847814961561600t^{22} + 13919661635344132997120t^{21} + 16946617223186869125120t^{20} + 15128485301432675729408t^{19} + 10569267773664849821696t^{18} + 6326120170068239187968t^{17} + 3545593211023962726400t^{16} + 1881655698676147400704t^{15} + 881906373612505790976t^{14} + 339256559767491995904t^{13} + 102684457275060186528t^{12} + 23930721826577894096t^{11} + 4222404410835383542t^{10} + 549417394490662055t^{9} + 49654018746843600t^{8} + 2601814934837422t^{7} + 11314286959820t^{6} - 7613970022088t^{5} - 304795910008t^{4} + 7326735706t^{3} + 584331790t^{2} + 6295401t - 8584 \right)D^{3} + 2 t \left( 424648474309007769600t^{24} + 2431518003968679608320t^{23} + 6611053784691187384320t^{22} + 11173417842538386030592t^{21} + 12977616410416156508160t^{20} + 11047646111087204499456t^{19} + 7451433559314063949824t^{18} + 4439814652169912713216t^{17} + 2541692678864398835712t^{16} + 1363574944262842916864t^{15} + 626546982078220198400t^{14} + 230432486178641454848t^{13} + 65681659081853261728t^{12} + 14288309311129804544t^{11} + 2339042491281295394t^{10} + 280478329078742013t^{9} + 23019248373650291t^{8} + 1041973613157452t^{7} - 2292056146497t^{6} - 2326102468570t^{5} + 13842647351t^{4} + 8914064724t^{3} + 318539951t^{2} + 1434205t - 1258 \right)D^{2} + 4 t ^{2} \left( 195901367982332313600t^{23} + 1084072098581792686080t^{22} + 2854779211389484400640t^{21} + 4668350460525270794240t^{20} + 5228788777077272739840t^{19} + 4292310181397865168896t^{18} + 2827651831406345584640t^{17} + 1691746634568067186688t^{16} + 987612843893937848320t^{15} + 531520869225299890176t^{14} + 238802313004621304832t^{13} + 84358143720952590336t^{12} + 22864185455563516992t^{11} + 4703788861467004688t^{10} + 726075754571898948t^{9} + 81913863692331742t^{8} + 6304707399080655t^{7} + 270162063409846t^{6} + 1659255317773t^{5} - 42989647270t^{4} + 46479942905t^{3} + 3395873826t^{2} + 87469191t + 132192 \right)D + 384 t ^{3} \left( 733163802329088000t^{22} + 3959700946283397120t^{21} + 10192150876315975680t^{20} + 16272860633991479296t^{19} + 17741875563834900480t^{18} + 14178272676530618368t^{17} + 9198670613180399616t^{16} + 5544802154993336320t^{15} + 3287822492165309440t^{14} + 1767868851397739520t^{13} + 778084366503702464t^{12} + 265865552511455552t^{11} + 69211836795058572t^{10} + 13628809647888656t^{9} + 2011988704950979t^{8} + 217392767527193t^{7} + 16132397097520t^{6} + 707237515300t^{5} + 16320239139t^{4} + 1591348035t^{3} + 209915550t^{2} + 9936816t + 203796 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 4, 6)

Show multiplicities

Sector dimension: 10
Spectral radius: $15.8499159$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$15.84992$	1	1
$-9.196467 - 2.4177\mathrm{i}$	1	1
$-9.196467 + 2.4177\mathrm{i}$	1	1
$-5$	1	1
$-2$	1	1
$-1.228491 - 1.161397\mathrm{i}$	1	1
$-1.228491 + 1.161397\mathrm{i}$	1	1
$-1$	2	2
$0$	1	1

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right)^{2} \left(\lambda + 2\right) \left(\lambda + 5\right) \left(\lambda^{5} + 5 \lambda^{4} - 192 \lambda^{3} - 1920 \lambda^{2} - 4096 \lambda - 4096\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 12 & 12 & 48 & 24 & 36 & 24 & 168 & 232 & 240 \\ 1 & -2 & 3 & 12 & 16 & 8 & 4 & 54 & 66 & 48 \\ 1 & 2 & -1 & 20 & 8 & 12 & 8 & 18 & 18 & 64 \\ 0 & 1 & 1 & -3 & 0 & 1 & 1 & 12 & 24 & 12 \\ 0 & 1 & 0 & 3 & -2 & 2 & 1 & 0 & 0 & 12 \\ 0 & 0 & 1 & 2 & 2 & -1 & 0 & 12 & 8 & 0 \\ 0 & 0 & 1 & 2 & 2 & -1 & -2 & 0 & -4 & 0 \\ 0 & 0 & 0 & \frac{5}{3} & -\frac{2}{3} & 2 & \frac{4}{3} & -3 & \frac{4}{3} & 8 \\ 0 & 0 & 0 & 1 & 2 & 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 4 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta