Family 2-131-33-a

← 2-132-34-a · 2-130-33-a →

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

$\pi_1\colon X\to W$ is the blow-up of the del Pezzo fourfold $W$ of degree $5$, a codimension-two linear section of $\mathrm{Gr}(2,5)$ (1-405-85-a), along a smooth rational surface of degree $10$, the blow-up of $\mathbb P^2$ in $9$ points.
$\pi_2\colon X\to Q^4$ maps birationally onto the smooth quadric fourfold (1-512-105-a) with center a surface of degree $8$, smooth except at one non-normal point; the fibers are lines over the smooth points and a quadric surface over the special point.

More details (pdf)

Varieties in this family can be blown down to

ℙ⁵P5 × Gr(2, 5) · 4150.wL11w3 (preferred model)

Defining data

Ambient: ℙ⁵P5 × Gr(2, 5)
Bundle: $(E^{ω_{7}})^{\oplus 2} \oplus E^{ω_{1} + ω_{6}} \oplus E^{ω_{1} + ω_{9}}$ 2 copies of E^(w7) + E^(w1 + w6) + E^(w1 + w9)
Bundle: $O_{2} (1)^{\oplus 2} \oplus O_{1} (1) ⊠ U_{2}^{\lor} \oplus O_{1} (1) ⊠ Q_{2}$ 2 copies of O_2(1) + O_1(1) box U_2^v + O_1(1) box Q_2
ZL62 label: 4150.wL11w3

Line bundles on the ambient $Y$

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

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	33
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	131
$\chi(X, \mathrm{T}_X)$	=	-13

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	131
$\mathrm{c}_1^2\mathrm{c}_2$	=	122
$\mathrm{c}_2^2$	=	115
$\mathrm{c}_1\mathrm{c}_3$	=	36
$\mathrm{c}_4$	=	18

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		12		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 + 13	0	x₀ + 13	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 29	0	0	x₁ + 14	15
$\mathrm{HH}^{ 4 }$ = 33	0	0	0	0	33
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 28t + 73t^{2} + 28t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(517)
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) \left( 2t + 1 \right) \left( 31099t^{8} + 65007t^{7} + 64741t^{6} + 39059t^{5} + 14121t^{4} + 2671t^{3} + 178t^{2} - 7t - 1 \right) \left( 1755183740322096t^{17} + 7381017957913632t^{16} + 14756615675358744t^{15} + 18869284600739536t^{14} + 17379350867637370t^{13} + 12135036789437732t^{12} + 6509645451363743t^{11} + 2662659161496068t^{10} + 815735245091325t^{9} + 182714222970551t^{8} + 29067680728960t^{7} + 3213399315364t^{6} + 262446311994t^{5} + 21840798218t^{4} + 2108819369t^{3} + 121836696t^{2} + 49663t + 7467 \right)D^{6} + \left( 2292547283891628267168t^{27} + 16763041424815014686544t^{26} + 58892769631979105787408t^{25} + 132852975774895490092536t^{24} + 216754692146974397969900t^{23} + 272703158408719097516654t^{22} + 274869036771290103577194t^{21} + 227157989872256030579119t^{20} + 155980555205526390353744t^{19} + 89568060885657441770875t^{18} + 43076942879042523605528t^{17} + 17315995661275249144498t^{16} + 5790116389330264164852t^{15} + 1599748702417137542877t^{14} + 362486519123418607065t^{13} + 66986991435330897558t^{12} + 10127077064157791552t^{11} + 1285050224740102293t^{10} + 145505569377348663t^{9} + 15636999251099040t^{8} + 1540110323494008t^{7} + 120444993245029t^{6} + 6390969903885t^{5} + 231449065835t^{4} + 10820857640t^{3} + 493794464t^{2} - 351t + 14934 \right)D^{5} + \left( 19104560699096902226400t^{27} + 132300526606832067479280t^{26} + 440420635040712380009712t^{25} + 942273695085521626468520t^{24} + 1459977526834633719342356t^{23} + 1746469670777773350671826t^{22} + 1674202734375334572611382t^{21} + 1313997328722837333128133t^{20} + 853814778158851095522364t^{19} + 461331218255510800873551t^{18} + 207251398258918844840727t^{17} + 77171620284006703550488t^{16} + 23691573847557539898728t^{15} + 5956420446893291285251t^{14} + 1218148603728216321993t^{13} + 202054859870809049868t^{12} + 27478986454034641031t^{11} + 3199376871435851520t^{10} + 345532491219383378t^{9} + 36066494344019813t^{8} + 3296949869488826t^{7} + 216076411924861t^{6} + 7506345225451t^{5} + 5500759173t^{4} - 8971839433t^{3} - 501001441t^{2} - 901715t - 7467 \right)D^{4} + t \left( 80239154936206989350880t^{26} + 531270310368827602214640t^{25} + 1692760790319053199745200t^{24} + 3472334949077974477481928t^{23} + 5168708294366836364607140t^{22} + 5949329092905367626408442t^{21} + 5488806264839527556768222t^{20} + 4138617564426111680313145t^{19} + 2573579893954790861154784t^{18} + 1323392706187322952140077t^{17} + 562107378516696033665970t^{16} + 196544410186011638420926t^{15} + 56315902821274789927092t^{14} + 13166884660542781944787t^{13} + 2511063220344480505799t^{12} + 395438201336155335710t^{11} + 53562771308597356678t^{10} + 6725162828920211133t^{9} + 823098950326026891t^{8} + 92252029292137294t^{7} + 8177096019187124t^{6} + 485136482742887t^{5} + 15342109230775t^{4} + 167204948133t^{3} + 2681996190t^{2} + 38945122t + 10887 \right)D^{3} + 2 t ^{2} \left( 88645161643809626330496t^{25} + 566310235764959984767104t^{24} + 1743067746011368960940832t^{23} + 3460260863269159255371112t^{22} + 4994720335030277434722280t^{21} + 5582640816058715886228472t^{20} + 5000958110619048429362984t^{19} + 3653949237792633913927521t^{18} + 2193412930438710384258634t^{17} + 1083240177024015744207916t^{16} + 439345577155969752750262t^{15} + 145862233065913183968850t^{14} + 39507674478028965480016t^{13} + 8723185904089308250535t^{12} + 1582647923498487845159t^{11} + 242957768879456794977t^{10} + 33594408856538673999t^{9} + 4477313042467254498t^{8} + 568722548450062392t^{7} + 61691769570171452t^{6} + 4969022861890218t^{5} + 254832754668191t^{4} + 6583899500259t^{3} + 65885535592t^{2} + 1753758277t + 13644436 \right)D^{2} + 8 t ^{2} \left( 24071746480862096805264t^{25} + 149679768698201146096872t^{24} + 448837476841559702226744t^{23} + 869385815023218799852944t^{22} + 1226486821395254422183870t^{21} + 1341160931795551782636687t^{20} + 1174994958932503459194467t^{19} + 837955988813160552009869t^{18} + 489277897235866348578611t^{17} + 233997536790721632262581t^{16} + 91462258463928533183001t^{15} + 29130968291097137426962t^{14} + 7546797091223875791382t^{13} + 1595716615009692602413t^{12} + 280318437488581783381t^{11} + 42848712278056658234t^{10} + 6130488659539888584t^{9} + 853721337840688859t^{8} + 108826805594093145t^{7} + 11228089083822276t^{6} + 824871232138404t^{5} + 37100881759492t^{4} + 804722970938t^{3} + 8951594799t^{2} + 243304257t + 1224588 \right)D + 24 t ^{2} \left( 3275067548416611810240t^{25} + 19987477785471598708320t^{24} + 58858625262027416218272t^{23} + 112074719357325133884912t^{22} + 155605764916399801495456t^{21} + 167564211496664162825532t^{20} + 144503059292496883460336t^{19} + 101262742317619592508476t^{18} + 57934320065196248235848t^{17} + 27051128823490380108159t^{16} + 10282994338791068775300t^{15} + 3173780123872385427661t^{14} + 795114116400238575280t^{13} + 162945470491990010839t^{12} + 28077213496247144524t^{11} + 4320656267657275808t^{10} + 638741873076611157t^{9} + 91291802234592929t^{8} + 11505531652694160t^{7} + 1130404723480929t^{6} + 76743619150866t^{5} + 3086802726697t^{4} + 58244515820t^{3} + 807135138t^{2} + 17807541t + 59736 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ⁵P5 × Gr(2, 5)

Sector dimension: 10
Spectral radius: $16.735552$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$16.73555$	1	1
$-7.907362 - 2.316997\mathrm{i}$	1	1
$-7.907362 + 2.316997\mathrm{i}$	1	1
$-2.375082$	1	1
$-2.050018 - 0.2554099\mathrm{i}$	1	1
$-2.050018 + 0.2554099\mathrm{i}$	1	1
$-2$	1	1
$-0.7228552 - 1.475676\mathrm{i}$	1	1
$-0.7228552 + 1.475676\mathrm{i}$	1	1
$-1$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda + 2\right) \left(\lambda^{8} + 7 \lambda^{7} - 178 \lambda^{6} - 2671 \lambda^{5} - 14121 \lambda^{4} - 39059 \lambda^{3} - 64741 \lambda^{2} - 65007 \lambda - 31099\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 12 & 12 & 66 & 42 & 27 & 42 & 144 & 296 & 340 \\ 1 & -1 & 3 & 12 & 12 & 2 & 4 & 36 & 66 & 64 \\ 1 & 2 & -1 & 20 & 12 & 8 & 12 & 24 & 48 & 80 \\ 0 & 1 & 1 & -2 & 0 & 1 & 1 & 8 & 20 & 12 \\ 0 & 1 & 0 & 3 & -1 & 1 & 2 & 0 & 0 & 12 \\ 0 & 0 & 1 & 2 & 2 & -2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 2 & 2 & 0 & -1 & 8 & 12 & 12 \\ 0 & 0 & 0 & \frac{5}{2} & -1 & \frac{11}{4} & \frac{33}{8} & -\frac{1}{2} & \frac{23}{4} & 12 \\ 0 & 0 & 0 & 1 & 2 & -\frac{1}{2} & -\frac{3}{4} & 1 & -\frac{3}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 4 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta