Family 2-116-30-a

← 2-118-30-a · 2-116-30-b →

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

$\varpi_1\colon X\to W$ is the blow-up of the del Pezzo fourfold $W$ of degree $5$, a linear section of $\mathrm{Gr}(2,5)$ (1-405-85-a), along a smooth rational surface of degree $11$, the blow-up of $\mathbb P^2$ in $10$ points.
$\varpi_3\colon X\to\mathbb P^4$ maps birationally onto $\mathbb P^4$ (1-625-126-a) with center a surface of degree $7$, smooth except at five non-normal points; the fibers are lines over the smooth points and quadric surfaces over the five special points.

More details (pdf)

Varieties in this family can be blown down to

Fl(1, 3, 5) · 44.22U (preferred model)

Defining data

Ambient: Fl(1, 3, 5)
Bundle: $(E^{ω_{3}})^{\oplus 2} \oplus E^{ω_{1} + ω_{4}}$ 2 copies of E^(w3) + E^(w1 + w4)
Bundle: $O (0, 1)^{\oplus 2} \oplus Q_{1}^{\lor} \otimes ⋀^{2} U_{2}^{\lor} \otimes Sym^{2} U_{3}^{\lor}$ 2 copies of O(0,1) + Q_1^v ot wedge^2 U_2^v ot Sym^2 U_3^v
ZL62 label: 44.22U

Line bundles on the ambient $Y$

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

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	30
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	116
$\chi(X, \mathrm{T}_X)$	=	-14

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	116
$\mathrm{c}_1^2\mathrm{c}_2$	=	116
$\mathrm{c}_2^2$	=	119
$\mathrm{c}_1\mathrm{c}_3$	=	34
$\mathrm{c}_4$	=	19

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		13		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 + 14	0	x₀ + 14	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 26	0	0	x₁ + 16	10
$\mathrm{HH}^{ 4 }$ = 30	0	0	0	0	30
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 25t + 64t^{2} + 25t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(552)
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( 2t + 1 \right) ^{2} \left( 3t + 1 \right) ^{3} \left( 944t^{5} + 736t^{4} + 1468t^{3} + 214t^{2} + 3t - 1 \right) \left( 144161840721920t^{15} + 507193720033280t^{14} + 849317289422848t^{13} + 870808865829888t^{12} + 589529480710144t^{11} + 269302369248512t^{10} + 83001236485312t^{9} + 17319466930944t^{8} + 2700388480672t^{7} + 454307010560t^{6} + 94527423824t^{5} + 15244773692t^{4} + 1365922586t^{3} + 48916459t^{2} - 90892t + 2987 \right)D^{6} + 2 \left( 2t + 1 \right) \left( 3t + 1 \right) ^{2} \left( 8573592991414026240t^{22} + 41162164104356986880t^{21} + 101911937663611977728t^{20} + 167096936131442999296t^{19} + 197909845494644121600t^{18} + 175832212037941168128t^{17} + 118773201567897953280t^{16} + 60992406738542179840t^{15} + 23667770439574811904t^{14} + 6920959076180204800t^{13} + 1559919678095029248t^{12} + 299751399117336160t^{11} + 59341401352130848t^{10} + 12498790217328864t^{9} + 2263821902371336t^{8} + 293154782386320t^{7} + 24531081060884t^{6} + 1265631068831t^{5} + 50565645969t^{4} + 2774154704t^{3} + 101130592t^{2} - 181143t + 2987 \right)D^{5} + \left( 3t + 1 \right) \left( 857359299141402624000t^{24} + 4604217063434767564800t^{23} + 12321161161871588065280t^{22} + 21498206433684934590464t^{21} + 27038393061216150749184t^{20} + 25718305989989181743104t^{19} + 18938254602901990203392t^{18} + 10894335703885451350016t^{17} + 4899861078549453244416t^{16} + 1720697124927521897984t^{15} + 475219837855099783680t^{14} + 107525132725997280896t^{13} + 22279960879660706496t^{12} + 4822170577652471936t^{11} + 1052338182759507616t^{10} + 195357052224093280t^{9} + 27057737127554152t^{8} + 2575103269545464t^{7} + 153524557160016t^{6} + 4593195795491t^{5} - 15297675759t^{4} - 6219823690t^{3} - 204401902t^{2} - 101569t - 2987 \right)D^{4} + 2 t \left( 5401363584590836531200t^{24} + 29686018151223086284800t^{23} + 80346608658124093194240t^{22} + 140646130698158656962560t^{21} + 176576881168844311445504t^{20} + 167312114262317372788736t^{19} + 122920894783898075369472t^{18} + 70952778938779039522816t^{17} + 32340009588354905909248t^{16} + 11667806570394052415744t^{15} + 3365692734146087894784t^{14} + 808149672503027512128t^{13} + 178722870560850708704t^{12} + 41005911773883055840t^{11} + 9554967649838448560t^{10} + 1950840007732880616t^{9} + 309005884914716688t^{8} + 35324803578579286t^{7} + 2737077810201953t^{6} + 130999224132580t^{5} + 3222635627439t^{4} + 28100752266t^{3} + 550082755t^{2} + 9561572t + 2781 \right)D^{3} + 4 t ^{2} \left( 5967220722024162263040t^{23} + 31848466776118891315200t^{22} + 83247536610832247586816t^{21} + 139885086600087024893952t^{20} + 167429269388209280032768t^{19} + 150135993029373503868928t^{18} + 103643308316583974629376t^{17} + 55862152736988673527808t^{16} + 23663273297123318060544t^{15} + 7925158720055825276928t^{14} + 2142116321046932356352t^{13} + 499929830313686767488t^{12} + 115195313956713317472t^{11} + 28462604779803933552t^{10} + 6790959689067746056t^{9} + 1337779961898817532t^{8} + 197895257434633754t^{7} + 20749113302236036t^{6} + 1449942679334996t^{5} + 61094070118865t^{4} + 1284291592217t^{3} + 12433565657t^{2} + 403673877t + 3261282 \right)D^{2} + 4 t ^{2} \left( 6481636301509003837440t^{23} + 33839806523219916226560t^{22} + 86228588613302533029888t^{21} + 140640517987700535754752t^{20} + 162470792297646842888192t^{19} + 139673075387848567848960t^{18} + 91779954914764952248320t^{17} + 46768505876316514973696t^{16} + 18630333011706748136448t^{15} + 5861740252950169488896t^{14} + 1506826910060182354432t^{13} + 349018838165173917184t^{12} + 84751007138226779200t^{11} + 22048715319296458400t^{10} + 5248063539711200272t^{9} + 990966100751751848t^{8} + 137812182370272564t^{7} + 13419658101037712t^{6} + 858982069487812t^{5} + 32464435193222t^{4} + 603950338593t^{3} + 7198794788t^{2} + 229719045t + 1147008 \right)D + 16 t ^{2} \left( 661391459337653452800t^{23} + 3401046868018131763200t^{22} + 8517823326598363545600t^{21} + 13613815382458678345728t^{20} + 15345519576073102917632t^{19} + 12802152733601411473408t^{18} + 8112794723434125381632t^{17} + 3961357081949374151680t^{16} + 1503851734295088986112t^{15} + 450358366787226498560t^{14} + 111612831939607130368t^{13} + 25989720552183962432t^{12} + 6626580413918078432t^{11} + 1777177333774574464t^{10} + 417417224078505904t^{9} + 75801945363214704t^{8} + 10013013296373832t^{7} + 917744134874218t^{6} + 54662643226483t^{5} + 1891192423996t^{4} + 32630083537t^{3} + 507405246t^{2} + 12703814t + 41818 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 5)

Show multiplicities

Sector dimension: 10
Spectral radius: $18.7298891$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$18.72989$	1	1
$-7.650441 - 3.852102\mathrm{i}$	1	1
$-7.650441 + 3.852102\mathrm{i}$	1	1
$-3$	3	1, 2
$-2$	2	2
$-0.2145033 - 0.8005915\mathrm{i}$	1	1
$-0.2145033 + 0.8005915\mathrm{i}$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 3\right)^{3} \left(\lambda^{5} - 3 \lambda^{4} - 214 \lambda^{3} - 1468 \lambda^{2} - 736 \lambda - 944\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 3\right)^{2} \left(\lambda^{5} - 3 \lambda^{4} - 214 \lambda^{3} - 1468 \lambda^{2} - 736 \lambda - 944\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 14 & 14 & 96 & 54 & 60 & 42 & 80 & 224 & 300 \\ 1 & -1 & 4 & 12 & 10 & 4 & 2 & 18 & 48 & 48 \\ 1 & 2 & -1 & 26 & 16 & 14 & 10 & 18 & 48 & 64 \\ 0 & 1 & 1 & -2 & 0 & 2 & 2 & 2 & 8 & 12 \\ 0 & 1 & 0 & 4 & -1 & 2 & 1 & 0 & 0 & 6 \\ 0 & 0 & 1 & 2 & 1 & -2 & -1 & 4 & 8 & 6 \\ 0 & 0 & 1 & 2 & 3 & 0 & -1 & 2 & 8 & 6 \\ 0 & 0 & 0 & -6 & -8 & 2 & 2 & -7 & -8 & 0 \\ 0 & 0 & 0 & \frac{25}{4} & \frac{21}{4} & \frac{7}{4} & 1 & \frac{7}{2} & 5 & 7 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & 4 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta