Family 2-162-39-a

← 2-163-39-a · 2-162-39-b →

Blow-up of the genus-eight Mukai fourfold along a smooth cubic scroll.

$\pi_1\colon X\to Y$ is the blow-up of a codimension-$4$ linear section $Y$ of $\mathrm{Gr}(2,6)$, a Fano fourfold of index $2$, degree $14$ and genus $8$ (1-224-51-a), along a cubic scroll $\cong\mathbb F_1$.
$\pi_2\colon X\to Q^4$ maps birationally onto a smooth quadric fourfold (1-512-105-a), with center a surface of degree $7$ and sectional genus $2$ having one improper double point; its normalization is a smooth cubic surface. The fibers are lines over the rest of the surface and a quadric surface over the double point.

More details (pdf)

Varieties in this family can be blown down to

Fl(2, 4, 6) · 59.33200 (preferred model)

Defining data

Ambient: Fl(2, 4, 6)
Bundle: $(E^{ω_{1}})^{\oplus 2} \oplus E^{ω_{3}} \oplus (E^{ω_{4}})^{\oplus 2}$ 2 copies of E^(w1) + E^(w3) + 2 copies of E^(w4)
Bundle: $(U_{3}^{\lor})^{\oplus 2} \oplus U_{2}^{\lor} \otimes ⋀^{2} U_{3}^{\lor} \oplus O (0, 1)^{\oplus 2}$ 2 copies of U_3^v + U_2^v ot wedge^2 U_3^v + 2 copies of O(0,1)
ZL62 label: 59.33200

Line bundles on the ambient $Y$

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

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	39
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	162
$\chi(X, \mathrm{T}_X)$	=	-10

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	162
$\mathrm{c}_1^2\mathrm{c}_2$	=	132
$\mathrm{c}_2^2$	=	110
$\mathrm{c}_1\mathrm{c}_3$	=	40
$\mathrm{c}_4$	=	16

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		10		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 + 10	0	x₀ + 10	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 34	0	0	x₁ + 8	26
$\mathrm{HH}^{ 4 }$ = 39	0	0	0	0	39
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 34t + 92t^{2} + 34t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

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( t + 1 \right) ^{2} \left( 20633t^{9} + 48522t^{8} + 65909t^{7} + 56176t^{6} + 29644t^{5} + 9936t^{4} + 1799t^{3} + 96t^{2} - 8t - 1 \right) \left( 321956299457856t^{21} + 3702545736937632t^{20} + 16172478248159052t^{19} + 38988400554118908t^{18} + 59940260497617183t^{17} + 62497526581134042t^{16} + 44129404145484033t^{15} + 18416690531985375t^{14} + 326398760282763t^{13} - 5669412199185027t^{12} - 4391169274873710t^{11} - 1839306894412707t^{10} - 461391749705508t^{9} - 59226935868527t^{8} + 1274073815725t^{7} + 2040046615191t^{6} + 435820040035t^{5} + 58952197993t^{4} + 6202365484t^{3} + 465932937t^{2} + 4485871t + 35559 \right)D^{6} + \left( t + 1 \right) \left( 139501410860992799808t^{31} + 2032613172991432310400t^{30} + 12394563810782998454964t^{29} + 44703603506743864038108t^{28} + 109856295115871342879547t^{27} + 197949421832944903899693t^{26} + 272231536586786578346904t^{25} + 290886237618470563026468t^{24} + 240441790450219240283820t^{23} + 147304219190737135855350t^{22} + 56206375206051109111026t^{21} - 1639391341268098526004t^{20} - 22395376002852274962315t^{19} - 20428520906791756512634t^{18} - 11745265029042547866001t^{17} - 4908306330844677420557t^{16} - 1524485245044690271797t^{15} - 339927516549256202479t^{14} - 46573410681774796410t^{13} - 289501663685415545t^{12} + 1752069203491062266t^{11} + 537019416389269280t^{10} + 104567232434444862t^{9} + 15832354239039447t^{8} + 1961664138052053t^{7} + 191722969323861t^{6} + 13300850706140t^{5} + 618150530261t^{4} + 31665828746t^{3} + 1823052689t^{2} + 12675315t + 71118 \right)D^{5} + \left( 1162511757174939998400t^{32} + 17839377380952517671360t^{31} + 114940769845732329557412t^{30} + 436950711260304077483388t^{29} + 1126786214436694917161877t^{28} + 2120149738952065062498486t^{27} + 3030518554822781614017333t^{26} + 3351289812857543786390511t^{25} + 2854014417551484749781192t^{24} + 1788099830409066403736271t^{23} + 679142390635952159856822t^{22} - 57778589185211246578926t^{21} - 332610471058743607913499t^{20} - 304727499293935294553734t^{19} - 179631220020521528800176t^{18} - 77339414489201550068541t^{17} - 24695997585848212721141t^{16} - 5564887938867329827461t^{15} - 696700721190832867523t^{14} + 51446551214626597364t^{13} + 53334900884760027011t^{12} + 16134338035941079923t^{11} + 3350479330724839210t^{10} + 545989784803962935t^{9} + 72155368781784744t^{8} + 7533223544476134t^{7} + 566421885447445t^{6} + 24580420005198t^{5} + 249190974703t^{4} - 19372870675t^{3} - 1776117979t^{2} - 12201465t - 35559 \right)D^{4} + t \left( 4882549380134747993280t^{31} + 74062179301202192004480t^{30} + 465474350842127416567260t^{29} + 1712418970207371731268192t^{28} + 4245946312251417067623645t^{27} + 7631955694195442348366724t^{26} + 10338045946113032272497219t^{25} + 10700698813205302532970216t^{24} + 8326077519201849031221144t^{23} + 4466982061948798888927182t^{22} + 999387410194105843374102t^{21} - 923408254831404515981394t^{20} - 1341377030912509080112245t^{19} - 974776176830557056632131t^{18} - 491116681782017239300949t^{17} - 180481722993567468415528t^{16} - 46796501967456396065356t^{15} - 7009938065188702370829t^{14} + 314682946158110713670t^{13} + 550113582359657768693t^{12} + 190426791638208899453t^{11} + 44179152127291296523t^{10} + 8083039364613781861t^{9} + 1234782356638214799t^{8} + 157139440930326444t^{7} + 15751625992848143t^{6} + 1115420628621316t^{5} + 46338726425121t^{4} + 881002525739t^{3} + 12850711964t^{2} + 98892825t + 29413 \right)D^{3} + t ^{2} \left( 10788109106583443185152t^{30} + 162182919869925673350816t^{29} + 999476273005232805513072t^{28} + 3582841015781829197796744t^{27} + 8613064240039274496151500t^{26} + 14933970972414164677285560t^{25} + 19384521254605060072583364t^{24} + 19013073731878945561305354t^{23} + 13679496459807254304681516t^{22} + 6264527218167024356398548t^{21} + 309735212069616030579906t^{20} - 2458489249072017346477644t^{19} - 2594058395402651908189464t^{18} - 1632729532202027594659876t^{17} - 722607724047658388797335t^{16} - 225971018746906710757379t^{15} - 43983235178125811445582t^{14} - 1158492874029670013146t^{13} + 2750403728352617677963t^{12} + 1201336394664884596915t^{11} + 322751843779517737648t^{10} + 66733832617866458024t^{9} + 11525609770149091391t^{8} + 1700377647742836841t^{7} + 207567867867203450t^{6} + 19380200409981038t^{5} + 1225187180726379t^{4} + 43493663565513t^{3} + 762748166716t^{2} + 12323454308t + 75901860 \right)D^{2} + 2 t ^{2} \left( 5859059256161697591936t^{30} + 87501333367899749608416t^{29} + 531333696224604835968408t^{28} + 1867795272963486622924560t^{27} + 4386449919817465848889038t^{26} + 7400835380650542659398308t^{25} + 9298029236638733550214158t^{24} + 8742460226512462003205886t^{23} + 5891709087105830780170980t^{22} + 2305426124070631800119874t^{21} - 332808752000646467257362t^{20} - 1365303882895820361440430t^{19} - 1223598978496967566849650t^{18} - 692760194864298026960090t^{17} - 272799526304333776577131t^{16} - 71113419001688274804505t^{15} - 8064119325436351546137t^{14} + 2567486252662591429865t^{13} + 1760475915665819668234t^{12} + 576481782994814729328t^{11} + 136928760823530706546t^{10} + 26583541968092049546t^{9} + 4446221583572324223t^{8} + 639307848999210541t^{7} + 74934367754132947t^{6} + 6525444878658067t^{5} + 370738768975586t^{4} + 11485174078832t^{3} + 199632207512t^{2} + 3039161282t + 14792544 \right)D + 4 t ^{2} \left( 1195726378808509712640t^{30} + 17777340232892492293440t^{29} + 106856659958657923106352t^{28} + 370575882721593929390616t^{27} + 856226617476392785228848t^{26} + 1417211777010686372533116t^{25} + 1739664277875047604331188t^{24} + 1586016517340220693091704t^{23} + 1016143000751224094815800t^{22} + 344131715423775109593006t^{21} - 119014914482433576126684t^{20} - 273831729455299588563474t^{19} - 222347325015906231901512t^{18} - 116242063877973834201852t^{17} - 41256933950069734637370t^{16} - 8717100251535303070242t^{15} - 25346446186473805803t^{14} + 798264106791236536017t^{13} + 363413509448626695949t^{12} + 104083911346725266693t^{11} + 23036999676550430940t^{10} + 4310984932502747156t^{9} + 707223002376469748t^{8} + 99633015996193720t^{7} + 11260094005885117t^{6} + 921552461629979t^{5} + 47667809322757t^{4} + 1322710003915t^{3} + 23841391190t^{2} + 314883286t + 1137888 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 4, 6)

Show multiplicities

Sector dimension: 11
Spectral radius: $13.6324607$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$13.63246$	1	1
$-7.14786 - 4.272931\mathrm{i}$	1	1
$-7.14786 + 4.272931\mathrm{i}$	1	1
$-2.855613$	1	1
$-1.094744 - 1.82057\mathrm{i}$	1	1
$-1.094744 + 1.82057\mathrm{i}$	1	1
$-1.481707$	1	1
$-0.4049658 - 0.9894124\mathrm{i}$	1	1
$-0.4049658 + 0.9894124\mathrm{i}$	1	1
$-1$	2	2

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda^{9} + 8 \lambda^{8} - 96 \lambda^{7} - 1799 \lambda^{6} - 9936 \lambda^{5} - 29644 \lambda^{4} - 56176 \lambda^{3} - 65909 \lambda^{2} - 48522 \lambda - 20633\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 8 & 12 & 42 & 9 & 39 & 21 & 60 & 28 & 35 \\ 1 & -1 & 3 & 2 & 2 & 2 & 0 & 0 & 27 & 12 & 4 \\ 1 & 1 & -1 & 6 & 16 & 4 & 12 & 6 & 9 & 3 & 8 \\ 0 & 1 & 0 & 0 & 3 & 0 & 3 & 1 & 0 & 0 & 3 \\ 0 & 1 & 1 & -1 & -2 & 0 & 1 & 1 & 6 & 4 & 3 \\ 0 & 1 & 0 & 0 & 3 & -1 & 2 & 1 & -6 & -4 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & -1 & 0 & 6 & 2 & 0 \\ 0 & 0 & 1 & 1 & 1 & 1 & -1 & -2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & \frac{26}{3} & \frac{4}{3} & 10 & \frac{17}{3} & 2 & 2 & \frac{16}{3} \\ 0 & 0 & 0 & 1 & -9 & 0 & -14 & -8 & -3 & -4 & -8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 4 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta