Family 2-146-36-c

← 2-146-36-b · 2-146-36-d →

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

$\varpi_3\colon X\to\mathbb P^4$ maps birationally onto $\mathbb P^4$ (1-625-126-a) with center a surface of degree $7$, smooth except at one non-normal point; the fibers are lines over the smooth points and a quadric surface over the special point.
$\varpi_1\colon X\to D$ maps birationally onto a fourfold of degree $19$ in a codimension-two linear section of $\mathrm{Gr}(2,6)$; it contracts two disjoint planes with normal bundle $\mathcal O(-1)^{\oplus 2}$ to the two singular points of the image, each locally the cone over $\mathbb P^1\times\mathbb P^2\subset\mathbb P^5$.

More details (pdf)

Varieties in this family can be blown down to

1-625-126-a

Fl(1, 3, 7) · 604.52211 (preferred model)

Defining data

Ambient: Fl(1, 3, 7)
Bundle: $(E^{ω_{2}})^{\oplus 2} \oplus (E^{ω_{3}})^{\oplus 2} \oplus E^{ω_{6}}$ 2 copies of E^(w2) + 2 copies of E^(w3) + E^(w6)
Bundle: $(U_{2}^{\lor} \otimes U_{3}^{\lor})^{\oplus 2} \oplus O (0, 1)^{\oplus 2} \oplus ⋀^{3} Q_{1}^{\lor} \otimes ⋀^{2} U_{2}^{\lor} \otimes U_{3}^{\lor}$ 2 copies of U_2^v ot U_3^v + 2 copies of O(0,1) + wedge^3 Q_1^v ot wedge^2 U_2^v ot U_3^v
ZL62 label: 604.52211

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,6) \\ \det\mathcal{E} &\cong \mathcal{O}(2,5) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	36
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	146
$\chi(X, \mathrm{T}_X)$	=	-16

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	146
$\mathrm{c}_1^2\mathrm{c}_2$	=	128
$\mathrm{c}_2^2$	=	115
$\mathrm{c}_1\mathrm{c}_3$	=	30
$\mathrm{c}_4$	=	21

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		15		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 + 16	0	x₀ + 16	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 32	0	0	x₁ + 8	24
$\mathrm{HH}^{ 4 }$ = 36	0	0	0	0	36
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 31t + 82t^{2} + 31t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

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( 2187t^{8} + 10935t^{7} + 20979t^{6} + 19615t^{5} + 9125t^{4} + 1906t^{3} + 83t^{2} - 4t - 1 \right) \left( 990763330536t^{17} + 12656569273488t^{16} + 62977097967516t^{15} + 169279179783804t^{14} + 295690977691020t^{13} + 374093015241654t^{12} + 352876764639069t^{11} + 231863352241914t^{10} + 88377894413532t^{9} + 8911413315939t^{8} - 5621381852651t^{7} - 1352956629700t^{6} + 374359465164t^{5} + 190402844606t^{4} + 27860311108t^{3} + 2083716270t^{2} + 14459136t + 175200 \right)D^{6} + \left( 36835589865997944t^{26} + 667248859522942704t^{25} + 5226573368314658988t^{24} + 24040366267829613480t^{23} + 74340223212920490684t^{22} + 167378747891423460228t^{21} + 288896806123279678737t^{20} + 393935081281392817746t^{19} + 428321796019429880070t^{18} + 367206485357387792808t^{17} + 240514079031367655409t^{16} + 113326062930087847263t^{15} + 33483887236361842413t^{14} + 3168196818453255255t^{13} - 1624709598474576577t^{12} - 470263305267792600t^{11} + 151739815267824093t^{10} + 127785763852620543t^{9} + 38388835131222955t^{8} + 6575435887915993t^{7} + 695821100217212t^{6} + 46447649666852t^{5} + 2226062794832t^{4} + 148497642260t^{3} + 8345614200t^{2} + 41625408t + 350400 \right)D^{5} + \left( 249181931446456680t^{26} + 4389291885374518752t^{25} + 32994257763073714164t^{24} + 144402579802854570996t^{23} + 423769075045750468332t^{22} + 907044153908988397626t^{21} + 1492598100881875778163t^{20} + 1938538596737675092608t^{19} + 1987414088110358077962t^{18} + 1569030894715253100975t^{17} + 904439041769294215857t^{16} + 338575086845092808901t^{15} + 50702062869622204149t^{14} - 20112741023199327809t^{13} - 12003218748820246536t^{12} - 897099439655802377t^{11} + 1322126738008868834t^{10} + 650628313699693822t^{9} + 158489613882338508t^{8} + 23946136398432987t^{7} + 2302607303641056t^{6} + 119240036300540t^{5} + 1199595223500t^{4} - 96909132478t^{3} - 7894685610t^{2} - 42417312t - 175200 \right)D^{4} + t \left( 855885764533481640t^{25} + 14759992683297988560t^{24} + 107443651014916805844t^{23} + 452091847830609304728t^{22} + 1273613760668928969972t^{21} + 2622406907818451457576t^{20} + 4158984188427596137503t^{19} + 5187793991307766579638t^{18} + 5039352067558315859802t^{17} + 3662965054791448081074t^{16} + 1829696631688514091735t^{15} + 486223067536726208769t^{14} - 51351461842402058457t^{13} - 96407057744303346159t^{12} - 30155897221218171833t^{11} + 1611940010100482306t^{10} + 4340203621230951817t^{9} + 1633585153883963951t^{8} + 341739748332544363t^{7} + 45952078080616839t^{6} + 3980270705300724t^{5} + 182809353324552t^{4} + 2873966379932t^{3} + 45245192748t^{2} + 381193536t + 148800 \right)D^{3} + 2 t ^{2} \left( 784381384205367984t^{24} + 13317580790870949936t^{23} + 94616372333533508676t^{22} + 386204011284501368568t^{21} + 1054512973709206118856t^{20} + 2108510376811998399576t^{19} + 3250159954120734418305t^{18} + 3920982873086308631220t^{17} + 3628512376593663407760t^{16} + 2433529111730488297920t^{15} + 1034901034885073959506t^{14} + 141977127467302931709t^{13} - 124663688059325202183t^{12} - 85138443468392605603t^{11} - 18520550368040730079t^{10} + 3477949817593338835t^{9} + 3480279086719273648t^{8} + 1108861754144514550t^{7} + 208824213251052020t^{6} + 25897071953292348t^{5} + 2054777144014332t^{4} + 83260016160384t^{3} + 1330585139052t^{2} + 24973150332t + 145387968 \right)D^{2} + 8 t ^{2} \left( 180927750224166372t^{24} + 3037885567910533872t^{23} + 21201854500656931212t^{22} + 84591885726792993060t^{21} + 225645847299838434384t^{20} + 441463019400264944250t^{19} + 665944080971124638601t^{18} + 781940748088196993493t^{17} + 693698368759817769189t^{16} + 431168075462264672844t^{15} + 152713847911957138089t^{14} - 3964929750279838071t^{13} - 35538121167525090672t^{12} - 17678067647885646759t^{11} - 2813974145623540200t^{10} + 1007010646576057340t^{9} + 700007178403735912t^{8} + 198454402156430932t^{7} + 34594853826313836t^{6} + 4031856498298124t^{5} + 293466872351189t^{4} + 10505873581807t^{3} + 183651316620t^{2} + 3315858096t + 13945920 \right)D + 48 t ^{2} \left( 10833997019411160t^{24} + 180524831365924416t^{23} + 1244221423378203960t^{22} + 4883665785349474080t^{21} + 12810534913235791890t^{20} + 24674908498151597820t^{19} + 36637150206032635032t^{18} + 42127057114670000502t^{17} + 36097149757747080726t^{16} + 20962529694429552972t^{15} + 6052883117270069547t^{14} - 1452998568777159420t^{13} - 2281503557034054894t^{12} - 953375871712832625t^{11} - 112505153105623806t^{10} + 65538494768464891t^{9} + 37369584280837331t^{8} + 9747064594130897t^{7} + 1605864998441391t^{6} + 178456300913287t^{5} + 11989357922080t^{4} + 382982833028t^{3} + 7541386500t^{2} + 118916880t + 350400 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 7)

Sector dimension: 10
Spectral radius: $14.4298701$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$14.42987$	1	1
$-6.381078 - 7.540147\mathrm{i}$	1	1
$-6.381078 + 7.540147\mathrm{i}$	1	1
$-1.730082 - 0.8562884\mathrm{i}$	1	1
$-1.730082 + 0.8562884\mathrm{i}$	1	1
$-1$	1	1
$-0.8322727 - 0.2738191\mathrm{i}$	1	1
$-0.8322727 + 0.2738191\mathrm{i}$	1	1
$-0.5430058$	1	1
$0$	1	1

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right) \left(\lambda^{8} + 4 \lambda^{7} - 83 \lambda^{6} - 1906 \lambda^{5} - 9125 \lambda^{4} - 19615 \lambda^{3} - 20979 \lambda^{2} - 10935 \lambda - 2187\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 6 & 6 & 66 & 33 & 36 & 69 & 120 & 64 & 180 \\ 1 & 0 & 4 & -8 & -2 & -4 & -10 & 54 & 30 & 32 \\ 1 & 0 & -1 & 18 & 12 & 8 & 16 & 63 & 33 & 32 \\ 0 & 1 & 1 & -1 & -2 & 1 & 1 & -12 & -6 & 12 \\ 0 & 1 & 0 & 4 & 1 & 2 & 5 & 0 & 0 & 12 \\ 0 & 0 & 1 & 0 & 1 & -2 & -1 & 12 & 6 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & -1 & 18 & 10 & 12 \\ 0 & 0 & 0 & -\frac{4}{3} & -\frac{14}{3} & \frac{4}{3} & 4 & -15 & -7 & 16 \\ 0 & 0 & 0 & 7 & 11 & 0 & -3 & 30 & 14 & -24 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{15}{4} & 2 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta