Family 2-277-61-a

← 2-288-63-a · 2-273-60-a →

Blow-up of the degree-$5$ del Pezzo fourfold along a smooth conic.

$\pi_1\colon X\to W$ is the blow-up of the del Pezzo fourfold $W$ of degree $5$, a linear section of $\mathrm{Gr}(2,5)$ of codimension $2$ (1-405-85-a), along a smooth conic.
$\pi_2\colon X\to\mathbb P^4$ resolves the projection of $W\subset\mathbb P^7$ (1-405-85-a) from the plane of the conic onto $\mathbb P^4$ (1-625-126-a); its center is a quartic surface of sectional genus $0$, the projection of a rational normal quartic scroll from a general point of $\mathbb P^5$, with one improper double point. The fibers are lines over the rest of the surface and a smooth quadric surface over the double point.

More details (pdf)

Varieties in this family can be blown up to

3-188-44-d

Varieties in this family can be blown down to

Fl(1, 3, 5) · 44.221 (preferred model)

Defining data

Ambient: Fl(1, 3, 5)
Bundle: $E^{ω_{2}} \oplus (E^{ω_{3}})^{\oplus 2}$ E^(w2) + 2 copies of E^(w3)
Bundle: $U_{2}^{\lor} \otimes U_{3}^{\lor} \oplus O (0, 1)^{\oplus 2}$ U_2^v ot U_3^v + 2 copies of O(0,1)
ZL62 label: 44.221

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,4) \\ \det\mathcal{E} &\cong \mathcal{O}(1,3) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	61
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	277
$\chi(X, \mathrm{T}_X)$	=	1

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	277
$\mathrm{c}_1^2\mathrm{c}_2$	=	166
$\mathrm{c}_2^2$	=	97
$\mathrm{c}_1\mathrm{c}_3$	=	52
$\mathrm{c}_4$	=	10

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		4		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 - 1	0	x₀ - 1	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 51	0	0	x₁ - 18	69
$\mathrm{HH}^{ 4 }$ = 61	0	0	0	0	61
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 56t + 163t^{2} + 56t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

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( 5t^{2} + 2t + 1 \right) \left( 23t^{2} + 6t - 1 \right) \left( 881t^{6} + 1167t^{5} + 1078t^{4} + 463t^{3} + 92t^{2} + 10t + 1 \right) \left( 24764886405804298t^{17} + 55969375761200176t^{16} + 48171212226693129t^{15} + 42739689145257522t^{14} + 77172216647117254t^{13} + 104899055762703835t^{12} + 87329866595824890t^{11} + 47333485636262236t^{10} + 17708058813028024t^{9} + 4829510508265160t^{8} + 1017677507634677t^{7} + 171652223652630t^{6} + 22442848580494t^{5} + 2091220834651t^{4} + 127926281648t^{3} + 4519524780t^{2} + 30312354t + 734050 \right)D^{6} + \left( 52690143790285311489270t^{27} + 209441219058137880921306t^{26} + 391884248057144021091061t^{25} + 514099460728978533583793t^{24} + 647542347923142491016980t^{23} + 885997995643113113131907t^{22} + 1135993662304461432757767t^{21} + 1195142494385308333649349t^{20} + 993253318023652828675859t^{19} + 653190385951607465833625t^{18} + 343744865515519087296631t^{17} + 146559113945885825216975t^{16} + 51237030672246663048931t^{15} + 14861438782408720803251t^{14} + 3617063608559452460745t^{13} + 745529813945464498599t^{12} + 130651066045475892699t^{11} + 19406473694600201965t^{10} + 2429751926046618082t^{9} + 258410990566370960t^{8} + 24391158819703773t^{7} + 2190221850874830t^{6} + 185981421573460t^{5} + 13039134430720t^{4} + 638359729800t^{3} + 17799330396t^{2} + 82128462t + 1468100 \right)D^{5} + \left( 439084531585710929077250t^{27} + 1650300959138709022821070t^{26} + 2836972640207966110653831t^{25} + 3498322321015326635838969t^{24} + 4637402960754769634095858t^{23} + 6897050265441695319659864t^{22} + 8962358594833265632148216t^{21} + 9067984665818166338459233t^{20} + 7079356920113522016836974t^{19} + 4333071852984823061481798t^{18} + 2116553549361153256697116t^{17} + 838729259265793093809342t^{16} + 273907567085419290779910t^{15} + 74823611823012147032374t^{14} + 17293153495401854462098t^{13} + 3391282318175741869248t^{12} + 558304842324104313320t^{11} + 75152847019981851432t^{10} + 7924775675121275745t^{9} + 608330321359019275t^{8} + 27447463881725320t^{7} - 311497481905150t^{6} - 167400651962434t^{5} - 13625363216769t^{4} - 604353950712t^{3} - 17105146956t^{2} - 67690668t - 734050 \right)D^{4} + t \left( 1844155032659985902124450t^{26} + 6617623669441525945822110t^{25} + 10618393143039267200854795t^{24} + 12617040047772268837764591t^{23} + 17950511532709093174114536t^{22} + 28402284083154144923929455t^{21} + 36696922126762526890602175t^{20} + 35623442518759761742550643t^{19} + 26304470624345200535080839t^{18} + 15151883845980951449289849t^{17} + 6962530586201235215364275t^{16} + 2602696647805875453901383t^{15} + 806195548155160537097485t^{14} + 210412501579687187202567t^{13} + 46787109214150029380571t^{12} + 8867590093636663553019t^{11} + 1415408321030150297889t^{10} + 186175255880733907061t^{9} + 19643646265495279386t^{8} + 1611625983277880318t^{7} + 98442683613139111t^{6} + 4121555329870366t^{5} + 99940347800786t^{4} + 1259545577118t^{3} + 20595921498t^{2} + 56907168t + 312700 \right)D^{3} + 2 t ^{2} \left( 2037352226557698710918440t^{25} + 7045797750468558563338316t^{24} + 10707721157279900158437999t^{23} + 12485114119827494519997257t^{22} + 19061921085735814771016348t^{21} + 31396848901933415275381229t^{20} + 40054112917603866096303253t^{19} + 37503918440326543417448415t^{18} + 26459301278630068467898061t^{17} + 14513498707666351772658588t^{16} + 6353204618912558954106743t^{15} + 2269633759555315105634106t^{14} + 675360646980353367645757t^{13} + 170258350191144000491546t^{12} + 36673503841984898289377t^{11} + 6719756201564388446196t^{10} + 1030544779957811466729t^{9} + 129318168663402789426t^{8} + 12951870232328415454t^{7} + 1006503722583137469t^{6} + 58236908693877203t^{5} + 2325839874090913t^{4} + 57224155438278t^{3} + 967764632605t^{2} + 15305070134t + 18017742 \right)D^{2} + 4 t ^{2} \left( 1106493019595991541274670t^{25} + 3721087552507260607245006t^{24} + 5428389642848467923592846t^{23} + 6281627057116451720113552t^{22} + 10176317446328256118618933t^{21} + 17175231672645530364803821t^{20} + 21631209748991700391831527t^{19} + 19693218938086071555602564t^{18} + 13419509816539869030468696t^{17} + 7091106091268949879196928t^{16} + 2991476863063937390375507t^{15} + 1032873924390691325474374t^{14} + 298349653492030009379300t^{13} + 73295395100506727937778t^{12} + 15396026590786560623641t^{11} + 2740586862923372686958t^{10} + 405478389627232508334t^{9} + 48733027300194161792t^{8} + 4648256071836173837t^{7} + 342466948372385636t^{6} + 18692092694051909t^{5} + 702547082360035t^{4} + 17003796961642t^{3} + 328701700932t^{2} + 4594214582t + 2348960 \right)D + 24 t ^{3} \left( 75271633986121873556100t^{24} + 248286098826389838000180t^{23} + 352056691647405571858376t^{22} + 406558048806494294747168t^{21} + 687599771990827103122943t^{20} + 1177021470827955083951913t^{19} + 1467744483669856922892199t^{18} + 1310407316794516987690670t^{17} + 871652854381898741394990t^{16} + 448667800853411787664335t^{15} + 184392758464221866929595t^{14} + 62152775612334730186373t^{13} + 17582516881082893510258t^{12} + 4240772584076786210457t^{11} + 874210442050938937071t^{10} + 152103701308347396531t^{9} + 21858450155559046914t^{8} + 2535421731517039675t^{7} + 232119669695472247t^{6} + 16327036443722497t^{5} + 844873340236269t^{4} + 29946796391248t^{3} + 711449505400t^{2} + 14884299081t + 178374150 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 5)

Sector dimension: 10
Spectral radius: $8.65685425$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$8.656854$	1	1
$-1.345343 - 7.194386\mathrm{i}$	1	1
$-1.345343 + 7.194386\mathrm{i}$	1	1
$-3.122877 - 0.90108\mathrm{i}$	1	1
$-3.122877 + 0.90108\mathrm{i}$	1	1
$-2.656854$	1	1
$-1 - 2\mathrm{i}$	1	1
$-1 + 2\mathrm{i}$	1	1
$-0.5317796 - 1.128701\mathrm{i}$	1	1
$-0.5317796 + 1.128701\mathrm{i}$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda^{2} + 2 \lambda + 5\right) \left(\lambda^{2} - 6 \lambda - 23\right) \left(\lambda^{6} + 10 \lambda^{5} + 92 \lambda^{4} + 463 \lambda^{3} + 1078 \lambda^{2} + 1167 \lambda + 881\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 9 & 9 & 6 & 3 & 8 & 16 & 5 \\ 2 & 0 & 3 & 0 & -2 & 0 & 0 & 0 & 0 & 4 \\ 1 & 0 & -1 & 0 & 2 & 0 & 0 & 3 & 9 & 4 \\ 0 & 1 & 2 & -1 & 0 & 1 & 1 & 0 & -2 & 0 \\ 0 & 2 & 0 & 3 & 0 & 2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 2 & 3 \\ 0 & 0 & 1 & 0 & 0 & -1 & -2 & 0 & 2 & 0 \\ 0 & 0 & 0 & 9 & 2 & 9 & 6 & -1 & 0 & 0 \\ 0 & 0 & 0 & 2 & 3 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 5 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta