Family 2-147-36-a

← 2-157-38-a · 2-147-36-b →

Blow-up of the degree-$5$ del Pezzo fourfold along a smooth rational surface of degree $9$.

$\pi_1\colon X\to W$ 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 $9$, sectional genus $3$ and $K^2=2$.
$\pi_2\colon X\to Z$ is the blow-up of a smooth complete intersection $Z$ of two quadrics in $\mathbb P^6$ (1-324-70-a) along a quintic rational normal scroll $S(2,3)\cong\mathbb F_1$.

More details (pdf)

Varieties in this family can be blown down to

ℙ⁶P6 × Gr(2, 5) · 41600.wOwO11 (preferred model)

Defining data

Ambient: ℙ⁶P6 × Gr(2, 5)
Bundle: $(E^{ω_{8}})^{\oplus 2} \oplus (E^{ω_{1} + ω_{10}})^{\oplus 2}$ 2 copies of E^(w8) + 2 copies of E^(w1 + w10)
Bundle: $O_{2} (1)^{\oplus 2} \oplus (O_{1} (1) ⊠ Q_{2})^{\oplus 2}$ 2 copies of O_2(1) + 2 copies of O_1(1) box Q_2
ZL62 label: 41600.wOwO11

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(7;5) \\ \det\mathcal{E} &\cong \mathcal{O}(6;4) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	36
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	147
$\chi(X, \mathrm{T}_X)$	=	-10

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	147
$\mathrm{c}_1^2\mathrm{c}_2$	=	126
$\mathrm{c}_2^2$	=	113
$\mathrm{c}_1\mathrm{c}_3$	=	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 + 31	0	0	x₁ + 11	20
$\mathrm{HH}^{ 4 }$ = 36	0	0	0	0	36
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 31t + 83t^{2} + 31t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

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) \left( 7825t^{9} + 33005t^{8} + 53598t^{7} + 46111t^{6} + 24821t^{5} + 8671t^{4} + 1893t^{3} + 147t^{2} - 7t - 1 \right) \left( 106714955194200t^{16} + 500436958687080t^{15} + 632336261147880t^{14} - 430367318848192t^{13} - 2235908645352713t^{12} - 3174630883912576t^{11} - 2639956334024277t^{10} - 1493822117583881t^{9} - 621232859323549t^{8} - 196040986143468t^{7} - 46232403223237t^{6} - 7731155928711t^{5} - 830770531343t^{4} - 47795251124t^{3} - 869713403t^{2} - 45226921t + 438648 \right)D^{6} + \left( 17535935012286915000t^{26} + 160222139987826984000t^{25} + 601739715458455212000t^{24} + 1112239279102607449560t^{23} + 569423890190765275975t^{22} - 2249249575452500309326t^{21} - 6872048002613008471994t^{20} - 10846238888877838000567t^{19} - 11878286243764901815330t^{18} - 9849336767411308921805t^{17} - 6450928026965081769374t^{16} - 3422708258762184683415t^{15} - 1494638754983291164020t^{14} - 541860824584396013297t^{13} - 163376238390600548805t^{12} - 40746186330292488285t^{11} - 8303545543225630712t^{10} - 1358982781642368965t^{9} - 175533430259235747t^{8} - 17832649648572900t^{7} - 1489456985987333t^{6} - 113073493832936t^{5} - 7623260310229t^{4} - 311002712448t^{3} - 4041513415t^{2} - 184416868t + 1315944 \right)D^{5} + \left( 146132791769057625000t^{26} + 1245099384694622580000t^{25} + 4233758707195591406400t^{24} + 6485602881365715543920t^{23} - 351466543527085881095t^{22} - 22597729710420219458458t^{21} - 53175543215533661066542t^{20} - 74326972541780308457748t^{19} - 74211781923973168234024t^{18} - 56689617279613159884578t^{17} - 34386347064010906017353t^{16} - 16949242552529694632850t^{15} - 6879279322845666116228t^{14} - 2308400575958598597444t^{13} - 637126431262862681742t^{12} - 142614247180377327208t^{11} - 25289110497298811598t^{10} - 3426531008984856687t^{9} - 333633486819246678t^{8} - 20251940661254650t^{7} - 327430451500615t^{6} + 64727407162809t^{5} + 7026771148455t^{4} + 389043620170t^{3} + 4367727153t^{2} + 162849635t - 877296 \right)D^{4} + t \left( 613757725430042025000t^{25} + 4932136535544427290000t^{24} + 15342453203131954284000t^{23} + 19258105050212513625000t^{22} - 14118168002048068358315t^{21} - 100118813071874591271626t^{20} - 203473618148439503990322t^{19} - 261317110959418309569961t^{18} - 243765746207506669410098t^{17} - 175286986622209506155911t^{16} - 100576302908071612585450t^{15} - 47043340248890343276029t^{14} - 18125457211810822870850t^{13} - 5751422887964498126315t^{12} - 1488121358321115049861t^{11} - 308267470068367626795t^{10} - 49803420949568565940t^{9} - 6057369790862560607t^{8} - 530137734958753619t^{7} - 32016279837005468t^{6} - 1400872269116867t^{5} - 62500271600068t^{4} - 2437197699997t^{3} + 39355314240t^{2} + 918934661t - 1497968 \right)D^{3} + 2 t \left( 678056153808427380000t^{25} + 5197563428645723056500t^{24} + 14972945481611934342000t^{23} + 15252264167849296292680t^{22} - 23956903499127101752420t^{21} - 111322989625467543344440t^{20} - 205806445129225087375149t^{19} - 248580445852537126431258t^{18} - 220399037181079742228965t^{17} - 151484886891326184578470t^{16} - 83414106445181932153625t^{15} - 37524720812111555945138t^{14} - 13892476632932223357297t^{13} - 4214962704977161736916t^{12} - 1033601506221944051151t^{11} - 200500187902681449517t^{10} - 29893518562303453292t^{9} - 3301733496786402716t^{8} - 259649367959966984t^{7} - 14608165735293073t^{6} - 713964784336489t^{5} - 35446657024184t^{4} - 709300325304t^{3} + 30758621805t^{2} + 175901817t - 219324 \right)D^{2} + 4 t ^{2} \left( 368254635258025215000t^{24} + 2722808493916325371500t^{23} + 7368545117736324702000t^{22} + 6081648372583006262760t^{21} - 15744295875621088551485t^{20} - 59889892790131672997310t^{19} - 103872327005818012932613t^{18} - 120005175607928311844107t^{17} - 102471329036210041910009t^{16} - 68107784946023483532573t^{15} - 36374289400579978783009t^{14} - 15889365562609372156689t^{13} - 5701954893462599062137t^{12} - 1668309358861143808279t^{11} - 391455721096704501062t^{10} - 71969434973688357748t^{9} - 10068894610505601570t^{8} - 1036515443676592311t^{7} - 76840401436620759t^{6} - 4413936465409747t^{5} - 250192387119778t^{4} - 11587233172334t^{3} - 31893377861t^{2} + 9803754857t + 12992448 \right)D + 48 t ^{3} \left( 12525667865919225000t^{23} + 90314444133107947500t^{22} + 233418771074685193200t^{21} + 158966169873115030860t^{20} - 590371166269201378115t^{19} - 2013180017017159382166t^{18} - 3348396893314217312737t^{17} - 3751612627058706631481t^{16} - 3119747906422627643741t^{15} - 2024822200694991651761t^{14} - 1058049647002258339161t^{13} - 452389166144768817820t^{12} - 158570260921606337910t^{11} - 45106940366764475812t^{10} - 10222400084437086095t^{9} - 1801506834918175506t^{8} - 239958705291175214t^{7} - 23512638418140932t^{6} - 1704435973079365t^{5} - 104720485425685t^{4} - 6562735070996t^{3} - 269560874604t^{2} + 2229067224t + 195417684 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ⁶P6 × Gr(2, 5)

Sector dimension: 10
Spectral radius: $14.8104546$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$14.81045$	1	1
$-8.061981 - 0.767509\mathrm{i}$	1	1
$-8.061981 + 0.767509\mathrm{i}$	1	1
$-0.677711 - 2.169251\mathrm{i}$	1	1
$-0.677711 + 2.169251\mathrm{i}$	1	1
$-1.425943 - 0.9680503\mathrm{i}$	1	1
$-1.425943 + 0.9680503\mathrm{i}$	1	1
$-1$	1	1
$-0.8876203$	1	1
$-0.5915634$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda^{9} + 7 \lambda^{8} - 147 \lambda^{7} - 1893 \lambda^{6} - 8671 \lambda^{5} - 24821 \lambda^{4} - 46111 \lambda^{3} - 53598 \lambda^{2} - 33005 \lambda - 7825\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 10 & 10 & 42 & 33 & 15 & 27 & 96 & 156 & 180 \\ 1 & -1 & 2 & 12 & 14 & 2 & 4 & 24 & 42 & 48 \\ 1 & 2 & -1 & 14 & 8 & 6 & 10 & 12 & 18 & 48 \\ 0 & 1 & 1 & -2 & 0 & 0 & 0 & 8 & 12 & 0 \\ 0 & 1 & 0 & 2 & -1 & 1 & 2 & 0 & 0 & 12 \\ 0 & 0 & 1 & 2 & 1 & -1 & -2 & -8 & -12 & 0 \\ 0 & 0 & 1 & 2 & 3 & 0 & 0 & 8 & 14 & 12 \\ 0 & 0 & 0 & \frac{33}{4} & \frac{61}{4} & -\frac{3}{2} & -\frac{15}{4} & 1 & \frac{13}{2} & 10 \\ 0 & 0 & 0 & -2 & -7 & 2 & 4 & 0 & -3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{13}{4} & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta