Family 2-224-51-b

← 2-224-51-a · 2-224-51-c →

Blow-up of $\mathbb P^4$ along a non-normal projected del Pezzo surface of degree $5$.

$\pi_1\colon X\to Q^3$, onto a smooth quadric threefold, is a fibration whose fibers are lines, except over $5$ points, where they are planes.
$\pi_2\colon X\to\mathbb P^4$ is the blow-up of $\mathbb P^4$ (1-625-126-a) along a quintic surface of sectional genus $1$, the projection of a quintic del Pezzo surface from a general point of $\mathbb P^5$, with one improper double point; the fibers are lines over the rest of the surface and a quadric surface over the double point.

More details (pdf)

Varieties in this family can be blown down to

1-625-126-a

Fl(1, 3, 5) · 44.32T (preferred model)

Defining data

Ambient: Fl(1, 3, 5)
Bundle: $E^{ω_{3}} \oplus E^{ω_{4}} \oplus E^{ω_{1} + ω_{3}}$ E^(w3) + E^(w4) + E^(w1 + w3)
Bundle: $O (0, 1) \oplus Q_{1}^{\lor} \otimes ⋀^{2} U_{2}^{\lor} \otimes U_{3}^{\lor} \oplus O (1, 1)$ O(0,1) + Q_1^v ot wedge^2 U_2^v ot U_3^v + O(1,1)
ZL62 label: 44.32T

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,4) \\ \det\mathcal{E} &\cong \mathcal{O}(1,3) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	51
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	224
$\chi(X, \mathrm{T}_X)$	=	-5

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	224
$\mathrm{c}_1^2\mathrm{c}_2$	=	152
$\mathrm{c}_2^2$	=	101
$\mathrm{c}_1\mathrm{c}_3$	=	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 + 44	0	0	x₁ - 5	49
$\mathrm{HH}^{ 4 }$ = 51	0	0	0	0	51
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 46t + 130t^{2} + 46t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(126)
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( 13631t^{8} + 8325t^{7} - 16431t^{6} - 16880t^{5} - 5634t^{4} - 694t^{3} - 11t^{2} + 4t + 1 \right) \left( 26448365279520t^{13} + 120949144287900t^{12} + 207121644258885t^{11} + 186985895498590t^{10} + 105044784554405t^{9} + 42574702386093t^{8} + 13924906489593t^{7} + 3612983444991t^{6} + 663454993923t^{5} + 79095479040t^{4} + 6046508664t^{3} + 284628036t^{2} + 1577506t + 39200 \right)D^{6} + \left( 7570871009627879520t^{22} + 46142376524655426840t^{21} + 109948321635842937645t^{20} + 125920021249801369020t^{19} + 49455300229172442125t^{18} - 53128343118992999847t^{17} - 97646553897392118603t^{16} - 80728594111001926427t^{15} - 45701555799381031228t^{14} - 20055090720467144410t^{13} - 7135733428860052346t^{12} - 2047029174058830847t^{11} - 460237638264874684t^{10} - 79120779921855266t^{9} - 10349595645728790t^{8} - 1061328889311480t^{7} - 92048562871697t^{6} - 7313076384829t^{5} - 526257084538t^{4} - 30564116030t^{3} - 1132271014t^{2} - 4340518t - 78400 \right)D^{5} + \left( 63090591746898996000t^{22} + 380025514728660559800t^{21} + 904652411556466559475t^{20} + 1103511958367503182470t^{19} + 677738021555661885075t^{18} + 55213098406002536880t^{17} - 267369684035278105285t^{16} - 269906375689677415180t^{15} - 163662733280672485363t^{14} - 75525838626280905800t^{13} - 27797115710955038559t^{12} - 8001296975914494894t^{11} - 1745798187502051483t^{10} - 282615918075670780t^{9} - 33409802097496254t^{8} - 2761023706987279t^{7} - 130449950362492t^{6} + 560095772663t^{5} + 468322486390t^{4} + 27579453258t^{3} + 1097522370t^{2} + 4407942t + 39200 \right)D^{4} + t \left( 264980485336975783200t^{21} + 1581276006036929960400t^{20} + 3742438642807646298075t^{19} + 4692043498451463434820t^{18} + 3356196671104534649075t^{17} + 1230001816088561997485t^{16} - 47948878838468980605t^{15} - 366040432138100276145t^{14} - 283776573133464153898t^{13} - 148459477168745193310t^{12} - 58501486107374483646t^{11} - 17222581666209080259t^{10} - 3732693373698157752t^{9} - 594311078670527416t^{8} - 69766216076577952t^{7} - 5943844612067832t^{6} - 337679236249947t^{5} - 10409573466235t^{4} - 149675483362t^{3} - 2858873850t^{2} - 26016354t - 22050 \right)D^{3} + 2 t ^{2} \left( 292740345705611341440t^{20} + 1734397634485268435490t^{19} + 4074536421143747285205t^{18} + 5160774660330096669370t^{17} + 3939105103014982198710t^{16} + 1876283232259414056751t^{15} + 511455673380843586272t^{14} - 5752032129749262638t^{13} - 105243130351881287678t^{12} - 76328865121294456686t^{11} - 34043742628292732253t^{10} - 10428956713879580714t^{9} - 2259988508106730160t^{8} - 354097971399823041t^{7} - 40727123677793711t^{6} - 3367797485904357t^{5} - 180307858607957t^{4} - 5107484624555t^{3} - 86748372402t^{2} - 1804788636t - 9984128 \right)D^{2} + 4 t ^{2} \left( 158988291202185469920t^{20} + 936968093308054483590t^{19} + 2185799051971981461720t^{18} + 2774825731176842749350t^{17} + 2181488051717648472275t^{16} + 1148016504596502056308t^{15} + 425370815335624221672t^{14} + 101921060064513443213t^{13} - 1931752745305347800t^{12} - 17887404129926335805t^{11} - 10185655167717411611t^{10} - 3341723892530129673t^{9} - 734007522490914663t^{8} - 114391509100073245t^{7} - 13002089297745686t^{6} - 1047266496153198t^{5} - 52688625368890t^{4} - 1371733495034t^{3} - 26965704920t^{2} - 515632880t - 1975680 \right)D + 16 t ^{2} \left( 16223295020631170400t^{20} + 95265038375046274050t^{19} + 221037092149470734250t^{18} + 280435639567342108050t^{17} + 223526371645530402075t^{16} + 123098673776478193410t^{15} + 50733566996370653340t^{14} + 15712519998431447525t^{13} + 2630248187611702755t^{12} - 600544935609223810t^{11} - 603786937215700545t^{10} - 219622283860484245t^{9} - 49496803456924557t^{8} - 7741166001780846t^{7} - 875938687960563t^{6} - 69104797795461t^{5} - 3286187084131t^{4} - 79973138007t^{3} - 1791854260t^{2} - 29433810t - 78400 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 5)

Sector dimension: 9
Spectral radius: $10.3219405$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$10.32194$	1	1
$-2.73059 - 7.359537\mathrm{i}$	1	1
$-2.73059 + 7.359537\mathrm{i}$	1	1
$-3.287419$	1	1
$-2.631062 - 0.8859387\mathrm{i}$	1	1
$-2.631062 + 0.8859387\mathrm{i}$	1	1
$-1.088378$	1	1
$-1$	1	1
$0.7771611$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda^{8} + 4 \lambda^{7} - 11 \lambda^{6} - 694 \lambda^{5} - 5634 \lambda^{4} - 16880 \lambda^{3} - 16431 \lambda^{2} + 8325 \lambda + 13631\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 0 & 15 & 15 & 3 & 20 & 8 & 5 \\ 2 & 0 & 3 & 0 & 0 & 0 & 3 & 0 & 8 \\ 1 & 0 & -1 & 2 & 4 & 0 & 9 & 3 & 4 \\ 0 & 1 & 2 & -1 & 0 & 1 & 0 & 0 & 3 \\ 0 & 2 & 0 & 3 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & -2 & 2 & 0 & 3 \\ 0 & 0 & 0 & 10 & 10 & 2 & 5 & 3 & 0 \\ 0 & 0 & 0 & -10 & -15 & 1 & -10 & -6 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 5 & 2 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta