Family 2-192-45-f

← 2-192-45-e · 2-192-45-g →

Blow-up of $\mathbb P^4$ along a Bordiga sextic surface.

$\pi_1\colon X\to\mathbb P^4$ is the blow-up of $\mathbb P^4$ (1-625-126-a) along a Bordiga surface, the blow-up of $\mathbb P^2$ in $10$ points embedded by quartics through them as a smooth surface of degree $6$ and sectional genus $3$ (the rank-two locus of a general $3\times4$ matrix of linear forms).
$\pi_2\colon X\to\mathbb P^3$ is a fibration whose fibers are lines (the trisecant lines of the Bordiga surface), except over $10$ points, where they are planes.

More details (pdf)

Varieties in this family can be blown up to

3-157-38-a

Varieties in this family can be blown down to

1-625-126-a

Fl(1, 4, 5) · 48.0UU (preferred model)

Defining data

Ambient: Fl(1, 4, 5)
Bundle: $E^{ω_{1}} \oplus (E^{ω_{1} + ω_{4}})^{\oplus 2}$ E^(w1) + 2 copies of E^(w1 + w4)
Bundle: $O (1, 0) \oplus O (1, 1)^{\oplus 2}$ O(1,0) + 2 copies of O(1,1)
ZL62 label: 48.0UU

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4,4) \\ \det\mathcal{E} &\cong \mathcal{O}(3,2) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	45
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	192
$\chi(X, \mathrm{T}_X)$	=	-12

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	192
$\mathrm{c}_1^2\mathrm{c}_2$	=	144
$\mathrm{c}_2^2$	=	106
$\mathrm{c}_1\mathrm{c}_3$	=	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 + 12	0	x₀ + 12	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 40	0	0	x₁	40
$\mathrm{HH}^{ 4 }$ = 45	0	0	0	0	45
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 40t + 110t^{2} + 40t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(132)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(175)

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( 7297t^{8} + 25300t^{7} + 33329t^{6} + 21156t^{5} + 6558t^{4} + 1000t^{3} + 13t^{2} - 3t - 1 \right) \left( 158931838777865t^{13} + 284395911356220t^{12} + 531623206149810t^{11} + 535647938549004t^{10} + 21042516124174t^{9} - 236852077951395t^{8} - 95706245361192t^{7} + 8732410175051t^{6} + 11256503900574t^{5} + 2280731342469t^{4} + 188875679430t^{3} + 14642676783t^{2} + 92649411t + 1540800 \right)D^{6} + \left( 24354238178803699005t^{22} + 133727132760574533365t^{21} + 377225209800142234335t^{20} + 734432545087613031387t^{19} + 997553349563833487110t^{18} + 839052469573269190004t^{17} + 289429410365714293449t^{16} - 174794168165359713838t^{15} - 257173178676153508670t^{14} - 123165200360540383045t^{13} - 16446819830035370532t^{12} + 10921950426715885638t^{11} + 6809276100518362883t^{10} + 1897710905940736419t^{9} + 319720160341422915t^{8} + 35557229830326977t^{7} + 2963399180341492t^{6} + 231243707466096t^{5} + 16033317653541t^{4} + 964619171700t^{3} + 58292557488t^{2} + 265621833t + 3081600 \right)D^{5} + \left( 202951984823364158375t^{22} + 990075655475441616175t^{21} + 2601815588784084744865t^{20} + 4835199546335689359065t^{19} + 6010614696931253265942t^{18} + 4055090058504867489195t^{17} + 198117168263360887262t^{16} - 2033562811134883741698t^{15} - 1671901275953986144106t^{14} - 536040773205536542313t^{13} + 31870712682788579462t^{12} + 93819110822753664007t^{11} + 38616619999038203297t^{10} + 8769302300927262641t^{9} + 1297374689393551647t^{8} + 136336680099417488t^{7} + 9928195119024163t^{6} + 330688309254705t^{5} - 11139388325358t^{4} - 772093442388t^{3} - 55209309129t^{2} - 259011657t - 1540800 \right)D^{4} + t \left( 852398336258129465175t^{21} + 3748071265150012453775t^{20} + 9362220242543062115125t^{19} + 16771032911509362697185t^{18} + 18958827931108471114850t^{17} + 9589859931305750274070t^{16} - 3655668341630610410015t^{15} - 8536063166216273096684t^{14} - 5143916386082901120828t^{13} - 1060128726147387341577t^{12} + 377035377641634648698t^{11} + 327250508912350791202t^{10} + 107963372616258101367t^{9} + 21249515530204721409t^{8} + 2816306956037742443t^{7} + 274575563268897593t^{6} + 19190620541932032t^{5} + 741304896715416t^{4} + 9758217979557t^{3} + 153992677860t^{2} + 1150910280t + 987075 \right)D^{3} + 4 t ^{2} \left( 470848604790204847430t^{20} + 1896985195052353273720t^{19} + 4578854011199299911695t^{18} + 7956802135103992774188t^{17} + 8164162232874105963981t^{16} + 2786481179234984240349t^{15} - 3167475120837462352636t^{14} - 4303878959041935642991t^{13} - 2065088058964567601305t^{12} - 225434780866880146639t^{11} + 240234478468396951041t^{10} + 143440753622735346884t^{9} + 40793534899282040645t^{8} + 7241301917815072887t^{7} + 895739440721682396t^{6} + 83367097973931846t^{5} + 5361519971834850t^{4} + 178859091057741t^{3} + 2641770363060t^{2} + 49115231316t + 222718464 \right)D^{2} + 4 t ^{2} \left( 511439001754877679105t^{20} + 1922510325575473797365t^{19} + 4539856325763772206185t^{18} + 7694741186553906167427t^{17} + 7205054608595340437775t^{16} + 1357810932213809920009t^{15} - 3990853430958464045120t^{14} - 4189593090692476644671t^{13} - 1671245139511671615250t^{12} - 35577399211315804815t^{11} + 257723353355517395915t^{10} + 125465511938885236928t^{9} + 32059076728493549198t^{8} + 5270362417221556419t^{7} + 622684183623338130t^{6} + 55714818602778456t^{5} + 3251372900056248t^{4} + 93564901897884t^{3} + 1597251446316t^{2} + 27832584780t + 85052160 \right)D + 8 t ^{2} \left( 104375306480587281450t^{20} + 373320915801727382850t^{19} + 869896424872943433570t^{18} + 1446931365284970838350t^{17} + 1255081692296182330291t^{16} + 73841349062886873105t^{15} - 862119973657143737927t^{14} - 777928100916346726541t^{13} - 267574497117206013657t^{12} + 15311837881262006323t^{11} + 50227797169973127844t^{10} + 21538307966713297925t^{9} + 5093812035944666720t^{8} + 791581468428528469t^{7} + 90702560035337567t^{6} + 7839197825985054t^{5} + 415519322957082t^{4} + 10489533044421t^{3} + 206968375566t^{2} + 3111000390t + 6163200 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 4, 5)

Sector dimension: 9
Spectral radius: $11.4729341$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$11.47293$	1	1
$-3.712951 - 7.970751\mathrm{i}$	1	1
$-3.712951 + 7.970751\mathrm{i}$	1	1
$-2.114615 - 2.046351\mathrm{i}$	1	1
$-2.114615 + 2.046351\mathrm{i}$	1	1
$-1.038879 - 0.4520998\mathrm{i}$	1	1
$-1.038879 + 0.4520998\mathrm{i}$	1	1
$-1$	1	1
$-0.7400442$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda^{8} + 3 \lambda^{7} - 13 \lambda^{6} - 1000 \lambda^{5} - 6558 \lambda^{4} - 21156 \lambda^{3} - 33329 \lambda^{2} - 25300 \lambda - 7297\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 2 & 21 & 12 & 21 & 48 & 40 & 10 \\ 1 & -1 & 0 & 2 & 0 & 6 & 27 & 21 & 4 \\ 2 & 3 & 0 & 0 & 0 & 0 & 27 & 15 & 12 \\ 0 & 2 & 1 & -1 & 0 & 0 & 0 & 0 & 3 \\ 0 & 1 & 0 & 0 & -1 & 0 & 6 & 2 & 3 \\ 0 & 0 & 2 & 3 & 3 & 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 2 & -\frac{1}{6} & \frac{11}{3} & 3 & 4 & -\frac{1}{3} \\ 0 & 0 & 0 & 1 & \frac{5}{2} & -2 & -3 & -4 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 9 & 7 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta