Family 2-163-39-a

← 2-163-40-a · 2-162-39-a →

Blow-up of $\mathbb P^4$ along an elliptic sextic scroll with three improper double points.

$\pi_1\colon X\to Y$ maps birationally onto a fourfold of degree $22$ in $\mathrm{Gr}(4,6)$; it contracts three disjoint planes with normal bundle $\mathcal O(-1)^{\oplus 2}$ to points.
$\pi_2\colon X\to\mathbb P^4$ maps birationally onto $\mathbb P^4$ (1-625-126-a) with center a sextic elliptic scroll, the projection of a normal elliptic scroll of degree $6$ in $\mathbb P^5$ with three improper double points, over which the fibers are quadric surfaces.

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Varieties in this family can be blown down to

1-625-126-a

Fl(1, 4, 6) · 58.220 (preferred model)

Defining data

Ambient: Fl(1, 4, 6)
Bundle: $E^{ω_{1}} \oplus (E^{ω_{3}})^{\oplus 2}$ E^(w1) + 2 copies of E^(w3)
Bundle: $O (1, 0) \oplus (⋀^{2} U_{2}^{\lor} \otimes U_{3}^{\lor})^{\oplus 2}$ O(1,0) + 2 copies of wedge^2 U_2^v ot U_3^v
ZL62 label: 58.220

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4,5) \\ \det\mathcal{E} &\cong \mathcal{O}(3,4) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	39
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	163
$\chi(X, \mathrm{T}_X)$	=	-6

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	163
$\mathrm{c}_1^2\mathrm{c}_2$	=	130
$\mathrm{c}_2^2$	=	109
$\mathrm{c}_1\mathrm{c}_3$	=	44
$\mathrm{c}_4$	=	8

Hodge diamond

				1
			0		0
		0		2		0
	0		1		1		0
0		0		6		0		0
	0		1		1		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 + 6	0	x₀ + 6	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 38	0	0	x₁ + 12	26
$\mathrm{HH}^{ 4 }$ = 39	0	0	0	0	39
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 34t + 93t^{2} + 34t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

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) ^{2} \left( 5t^{2} + 2t + 1 \right) ^{2} \left( 1589t^{4} + 1137t^{3} + 63t^{2} + t - 1 \right) \left( 1510391748270t^{14} - 884360269066t^{13} + 2689973238557t^{12} + 5535852337631t^{11} + 3781074026268t^{10} + 1384171278241t^{9} - 156742829474t^{8} - 437183544749t^{7} + 80937842740t^{6} + 262796037885t^{5} + 112191369931t^{4} + 18947867936t^{3} + 1271632252t^{2} + 7631962t + 130400 \right)D^{6} + \left( t + 1 \right) \left( 5t^{2} + 2t + 1 \right) \left( 252001311240108150t^{21} + 226595702905608950t^{20} + 430301980023869425t^{19} + 1729332394172225821t^{18} + 2788837064941675993t^{17} + 2425254716138469471t^{16} + 1228626771356547965t^{15} + 225633679182317532t^{14} - 117818253726272292t^{13} - 2594802203974762t^{12} + 128611659388464575t^{11} + 116253258209049775t^{10} + 53189406365101791t^{9} + 14963887481506751t^{8} + 2691486943834195t^{7} + 298010459514206t^{6} + 18912164647464t^{5} + 905803634782t^{4} + 87099481056t^{3} + 5044435826t^{2} + 21200686t + 260800 \right)D^{5} + \left( 10500054635004506250t^{24} + 18842458689892027000t^{23} + 29202020129236743915t^{22} + 99171355196361552766t^{21} + 209836078889832119255t^{20} + 258227042543281699497t^{19} + 203290027080948332494t^{18} + 99875560034471093856t^{17} + 21867489749311666514t^{16} - 2246817688825939113t^{15} + 5499209220522478626t^{14} + 13599025297794396342t^{13} + 12032022294272182476t^{12} + 6450124301430893241t^{11} + 2358776461726366634t^{10} + 609090374347327892t^{9} + 111151073729797816t^{8} + 14052791789752233t^{7} + 1166663098310203t^{6} + 53391763278460t^{5} + 311344762489t^{4} - 78312489674t^{3} - 4895744016t^{2} - 24028084t - 130400 \right)D^{4} + t \left( 44100229467018926250t^{23} + 61651278610948153000t^{22} + 103371782484724894775t^{21} + 406944381197534915960t^{20} + 816113953257242979475t^{19} + 919352394812648885136t^{18} + 645000931205436853342t^{17} + 256399188453283965937t^{16} + 15085495258092457218t^{15} - 19926316170042078791t^{14} + 34451891276711282561t^{13} + 62313460653371727854t^{12} + 47264822837383869665t^{11} + 22341978194891764600t^{10} + 7239668181557891862t^{9} + 1651222397062162469t^{8} + 263948310135456734t^{7} + 28828924651347633t^{6} + 2049040340404986t^{5} + 83440822528244t^{4} + 1235470307602t^{3} + 22122631926t^{2} + 207396298t + 105600 \right)D^{3} + 2 t ^{2} \left( 48720253506420909000t^{22} + 53329264942232202500t^{21} + 102826223934752508005t^{20} + 445433612543837157508t^{19} + 844916559443500471517t^{18} + 881270058432182209576t^{17} + 557168306807719719408t^{16} + 173917682001180411181t^{15} - 25602823036078377044t^{14} - 20261152863484648814t^{13} + 48585300619101759498t^{12} + 70842390698970828659t^{11} + 48067684083883994154t^{10} + 20671221294358858358t^{9} + 6099008781605843066t^{8} + 1259643086503587557t^{7} + 180505450827020090t^{6} + 17372508087092736t^{5} + 1069053968236843t^{4} + 38854726243303t^{3} + 618060776643t^{2} + 12995115804t + 77381316 \right)D^{2} + 4 t ^{2} \left( 26460137680211355750t^{22} + 23080615438592923300t^{21} + 52675089078240713700t^{20} + 241022726360095419940t^{19} + 435647204984821925487t^{18} + 426758714435495103830t^{17} + 246558278104344971444t^{16} + 58097735313991470104t^{15} - 25546922526790131524t^{14} - 8255771753744692394t^{13} + 30383980203607646873t^{12} + 38557953333693220684t^{11} + 24172157834283204101t^{10} + 9697115766901386112t^{9} + 2666628927661121846t^{8} + 510306456378265358t^{7} + 67061044408427894t^{6} + 5801657290314938t^{5} + 311457656012003t^{4} + 10167930236752t^{3} + 181045125018t^{2} + 3547946816t + 14396160 \right)D + 8 t ^{2} \left( 5400028098002317500t^{22} + 3899242117032027000t^{21} + 10426976095849905420t^{20} + 49113414654189148428t^{19} + 85640701861051515395t^{18} + 80089203128591229612t^{17} + 43077084974037768210t^{16} + 7412197906048808816t^{15} - 6429076469042060226t^{14} - 1168992186667282837t^{13} + 6719660359140011422t^{12} + 7820388811866257637t^{11} + 4647484332081156034t^{10} + 1776035859952409688t^{9} + 464439664838050740t^{8} + 84067876634306374t^{7} + 10354273469596448t^{6} + 823417260495843t^{5} + 39197114870234t^{4} + 1165252520255t^{3} + 23413584999t^{2} + 390344480t + 1043200 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 4, 6)

Show multiplicities

Sector dimension: 10
Spectral radius: $13.1163413$
Diagonalizable at 1: Yes
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$13.11634$	1	1
$-5.299482 - 7.19409\mathrm{i}$	1	1
$-5.299482 + 7.19409\mathrm{i}$	1	1
$-1 - 2\mathrm{i}$	2	1, 1
$-1 + 2\mathrm{i}$	2	1, 1
$-1.517378$	1	1
$-1$	2	1, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda^{2} + 2 \lambda + 5\right)^{2} \left(\lambda^{4} - \lambda^{3} - 63 \lambda^{2} - 1137 \lambda - 1589\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda^{2} + 2 \lambda + 5\right) \left(\lambda^{4} - \lambda^{3} - 63 \lambda^{2} - 1137 \lambda - 1589\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 54 & 27 & 57 & 33 & 104 & 64 & 35 \\ 1 & 0 & 4 & -8 & -4 & -8 & -4 & 30 & 18 & 4 \\ 1 & 0 & -1 & 14 & 10 & 12 & 6 & 48 & 30 & 8 \\ 0 & 1 & 1 & -1 & -2 & 1 & 1 & -10 & -6 & 0 \\ 0 & 1 & 0 & 4 & 1 & 4 & 2 & 0 & 0 & 3 \\ 0 & 0 & 1 & 0 & 1 & 0 & 1 & 18 & 10 & 6 \\ 0 & 0 & 1 & 0 & 1 & -2 & -3 & 2 & 2 & -3 \\ 0 & 0 & 0 & 5 & \frac{13}{2} & 0 & -1 & 14 & 10 & -2 \\ 0 & 0 & 0 & -4 & -\frac{17}{2} & 4 & 4 & -21 & -15 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 16 & 10 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta