Family 2-194-45-b

← 2-194-45-a · 2-193-45-a →

Blow-up of a smooth quadric fourfold along a non-normal sextic surface.

$\pi_1\colon X\to\mathrm{Gr}(5,7)$ is birational onto its image; it contracts two disjoint planes with normal bundle $\mathcal O(-1)^{\oplus 2}$ to points where the image is locally the cone over $\mathbb P^2\times\mathbb P^1$.
$\pi_2\colon X\to Q$ maps birationally onto a smooth quadric fourfold in $\mathbb P^5$ (1-512-105-a), with center a surface of degree $6$ and sectional genus $1$, the projection of a sextic del Pezzo surface from a point, 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 down to

1-512-105-a

Fl(1, 5, 7) · 60G.531 (preferred model)

Defining data

Ambient: Fl(1, 5, 7)
Bundle: $E^{ω_{2}} \oplus E^{ω_{4}} \oplus E^{ω_{6}}$ E^(w2) + E^(w4) + E^(w6)
Bundle: $U_{2}^{\lor} \otimes U_{3}^{\lor} \oplus ⋀^{3} U_{2}^{\lor} \otimes U_{3}^{\lor} \oplus Q_{1}^{\lor} \otimes ⋀^{4} U_{2}^{\lor} \otimes U_{3}^{\lor}$ U_2^v ot U_3^v + wedge^3 U_2^v ot U_3^v + Q_1^v ot wedge^4 U_2^v ot U_3^v
ZL62 label: 60G.531

Line bundles on the ambient $Y$

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

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	45
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	194
$\chi(X, \mathrm{T}_X)$	=	-5

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	194
$\mathrm{c}_1^2\mathrm{c}_2$	=	140
$\mathrm{c}_2^2$	=	107
$\mathrm{c}_1\mathrm{c}_3$	=	46
$\mathrm{c}_4$	=	13

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		7		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 + 5	0	x₀ + 5	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 38	0	0	x₁ + 1	37
$\mathrm{HH}^{ 4 }$ = 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

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

$\displaystyle\frac{ 1 + 40t + 112t^{2} + 40t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

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( 4t^{2} + 1 \right) \left( 1376t^{6} + 4992t^{5} + 2672t^{4} + 948t^{3} + 58t^{2} - 3t - 1 \right) \left( 1821472360448t^{17} - 5041888827392t^{16} + 8514983482368t^{15} + 52102295557632t^{14} + 86133001804288t^{13} + 81154662826368t^{12} + 46521012525664t^{11} + 26136627599616t^{10} + 13932712903928t^{9} + 3928778046544t^{8} + 112399347702t^{7} - 123794848470t^{6} - 25261027828t^{5} + 2398039900t^{4} + 258798619t^{3} + 37605865t^{2} + 320332t + 5238 \right)D^{6} + \left( 170431525822398464t^{26} + 103645061641928704t^{25} - 382406354457657344t^{24} + 7563678033590190080t^{23} + 33580872678938296320t^{22} + 68255601635905454080t^{21} + 87979933930067984384t^{20} + 80954294938099288064t^{19} + 59497869173149573120t^{18} + 37890388939248322048t^{17} + 20496770205518085888t^{16} + 8879121515834000768t^{15} + 2936199022166977728t^{14} + 659530323298521280t^{13} + 59588366813448288t^{12} - 14331648582683672t^{11} - 5886949571308356t^{10} - 597373478616186t^{9} + 69891651720040t^{8} + 30095778695322t^{7} + 2543512764174t^{6} + 196884728743t^{5} + 26230783336t^{4} + 1427533955t^{3} + 150744960t^{2} + 919092t + 10476 \right)D^{5} + \left( 1152919145269166080t^{26} - 408233141876490240t^{25} - 1276981523087032320t^{24} + 58230954686341758976t^{23} + 224091516342206980096t^{22} + 421640640828692586496t^{21} + 508823648185169502208t^{20} + 438377860176946125824t^{19} + 308688397694564554752t^{18} + 193383661471627065344t^{17} + 100609196605167973760t^{16} + 38309049204044745728t^{15} + 9438033816769556960t^{14} + 921774480107133168t^{13} - 318070432086958600t^{12} - 147128669063758680t^{11} - 26389982301003512t^{10} - 1218645240723094t^{9} + 364045491576942t^{8} + 109741063788350t^{7} + 14823646421246t^{6} + 1492604589777t^{5} + 16831513537t^{4} - 581476999t^{3} - 143167903t^{2} - 999602t - 5238 \right)D^{4} + t \left( 3960026629402787840t^{25} - 4293164674712207360t^{24} + 1357824827010187264t^{23} + 218604676681537716224t^{22} + 753431240891980005376t^{21} + 1337812820473656287232t^{20} + 1540513405568063827968t^{19} + 1274131010949320919040t^{18} + 886490030186577453056t^{17} + 557128488885865767424t^{16} + 278581715266735710976t^{15} + 92066484665792583424t^{14} + 15248136801503317120t^{13} - 1243079318671458528t^{12} - 1415313411119116496t^{11} - 358384897817191608t^{10} - 39988203984767556t^{9} + 2780902586652774t^{8} + 1309620447034528t^{7} + 197230221342014t^{6} + 21844887327094t^{5} + 1642451693107t^{4} + 32667752804t^{3} + 678465577t^{2} + 7117980t + 2754 \right)D^{3} + 2 t ^{2} \left( 3629188961629896704t^{24} - 5918773149628268544t^{23} + 6321333200436920320t^{22} + 212673207847801970688t^{21} + 675024289618887069696t^{20} + 1149608520671168290816t^{19} + 1280611451301457305600t^{18} + 1033021751352698571264t^{17} + 721027018917484922880t^{16} + 456288272593349145600t^{15} + 218645856077567071040t^{14} + 62425360337718912960t^{13} + 5523473862247555216t^{12} - 2750336426425202760t^{11} - 1205005398970981524t^{10} - 215077081629009888t^{9} - 14879040632854450t^{8} + 3342614698096082t^{7} + 826332452399242t^{6} + 105969291916062t^{5} + 12148550118453t^{4} + 778451313973t^{3} + 15189259544t^{2} + 375402601t + 2730453 \right)D^{2} + 8 t ^{2} \left( 837119553304133632t^{24} - 1702785372336406528t^{23} + 2457568370692284416t^{22} + 51093180060286296064t^{21} + 152626478062827681792t^{20} + 252362961598695634944t^{19} + 274603963334196529664t^{18} + 218225649001318646528t^{17} + 153702818320112011264t^{16} + 97858186961026791616t^{15} + 45055404199818112608t^{14} + 11187678461180194416t^{13} + 178311586084871824t^{12} - 784230796400124984t^{11} - 244119127898768600t^{10} - 32404634950264566t^{9} - 958596658687680t^{8} + 718090654511960t^{7} + 132870124498028t^{6} + 17010235989519t^{5} + 1907242623150t^{4} + 101928053857t^{3} + 2183742469t^{2} + 51866654t + 267138 \right)D + 16 t ^{2} \left( 150380758078586880t^{24} - 349041821155491840t^{23} + 580843480048656384t^{22} + 9434767990538055680t^{21} + 26993355547350073344t^{20} + 43731339095251636224t^{19} + 46817890195791434240t^{18} + 36884253322831973888t^{17} + 26234897639101198080t^{16} + 16775719142033401280t^{15} + 7471489918906526656t^{14} + 1640301037640525360t^{13} - 81879489431280792t^{12} - 151759385434494540t^{11} - 38823273028018522t^{10} - 3934510830123128t^{9} + 46213942387672t^{8} + 112022642978618t^{7} + 17674546308581t^{6} + 2463847931952t^{5} + 255967570550t^{4} + 11539935932t^{3} + 284532651t^{2} + 5919210t + 20952 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 5, 7)

Sector dimension: 10
Spectral radius: $11.4268281$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$11.42683$	1	1
$-5.724245 - 5.792754\mathrm{i}$	1	1
$-5.724245 + 5.792754\mathrm{i}$	1	1
$-1.326112 - 1.951651\mathrm{i}$	1	1
$-1.326112 + 1.951651\mathrm{i}$	1	1
$0 - 2\mathrm{i}$	1	1
$0 + 2\mathrm{i}$	1	1
$-1$	1	1
$-0.3261123$	1	1
$0$	1	1

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right) \left(\lambda^{2} + 4\right) \left(\lambda^{6} + 3 \lambda^{5} - 58 \lambda^{4} - 948 \lambda^{3} - 2672 \lambda^{2} - 4992 \lambda - 1376\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 36 & 18 & 42 & 18 & 48 & 48 & 120 \\ 1 & 0 & 3 & -6 & -2 & -8 & -2 & 36 & 36 & 0 \\ 1 & 0 & -1 & 12 & 8 & 12 & 4 & 24 & 18 & 24 \\ 0 & 1 & 1 & -1 & -2 & 0 & 1 & -8 & -6 & 0 \\ 0 & 1 & 0 & 3 & 1 & 4 & 1 & 0 & 0 & 18 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 16 & 18 & 18 \\ 0 & 0 & 1 & 0 & 1 & -2 & -2 & 0 & -6 & -18 \\ 0 & 0 & 0 & \frac{3}{2} & -\frac{1}{2} & 3 & \frac{3}{2} & -3 & -\frac{3}{2} & 6 \\ 0 & 0 & 0 & 1 & 2 & -\frac{2}{3} & -\frac{1}{3} & 4 & 2 & -4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{10}{3} & 3 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta