Family 2-273-60-a

← 2-277-61-a · 2-272-60-a →

Blow-up of $\mathbb P^4$ along a rational normal quartic curve.

$\pi_1\colon X\to\mathrm{Gr}(3,5)$ is the blow-up of a smooth quadric fourfold $Q^4$ (1-512-105-a) along a Veronese surface.
$\pi_2\colon X\to\mathbb P^4$ is the blow-up of $\mathbb P^4$ (1-625-126-a) along a rational normal quartic curve.

More details (pdf)

Varieties in this family can be blown up to

3-199-46-b

Varieties in this family can be blown down to

Fl(1, 3, 5) · 44.3U (preferred model)

Defining data

Ambient: Fl(1, 3, 5)
Bundle: $E^{ω_{4}} \oplus E^{ω_{1} + ω_{4}}$ E^(w4) + E^(w1 + w4)
Bundle: $Q_{1}^{\lor} \otimes ⋀^{2} U_{2}^{\lor} \otimes U_{3}^{\lor} \oplus Q_{1}^{\lor} \otimes ⋀^{2} U_{2}^{\lor} \otimes Sym^{2} U_{3}^{\lor}$ Q_1^v ot wedge^2 U_2^v ot U_3^v + Q_1^v ot wedge^2 U_2^v ot Sym^2 U_3^v
ZL62 label: 44.3U

Line bundles on the ambient $Y$

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

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	60
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	273
$\chi(X, \mathrm{T}_X)$	=	3

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	273
$\mathrm{c}_1^2\mathrm{c}_2$	=	162
$\mathrm{c}_2^2$	=	100
$\mathrm{c}_1\mathrm{c}_3$	=	54
$\mathrm{c}_4$	=	9

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		3		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 - 3	0	x₀ - 3	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 49	0	0	x₁ - 18	67
$\mathrm{HH}^{ 4 }$ = 60	0	0	0	0	60
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 55t + 161t^{2} + 55t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(29)
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( 4t^{2} + 1 \right) \left( 18772t^{7} + 13984t^{6} + 9717t^{5} + 2524t^{4} + 487t^{3} - 7t^{2} - 3t - 1 \right) \left( 757038771439104t^{13} + 851848681668816t^{12} + 483342631552864t^{11} + 311598988768112t^{10} + 120625673479736t^{9} - 4150598024948t^{8} - 6979532186192t^{7} - 1129941749644t^{6} - 135471700012t^{5} + 45777516454t^{4} + 1547894649t^{3} + 381165795t^{2} + 2560374t + 62244 \right)D^{6} + \left( 1193735072666208264192t^{22} + 2127076756027958490624t^{21} + 2259690758570586857984t^{20} + 1926698346884909244160t^{19} + 1242208023101465523840t^{18} + 610497814040821752416t^{17} + 231621490902566469984t^{16} + 54046029922815257568t^{15} + 4771650309091822136t^{14} - 2825191203100016200t^{13} - 1390899699469079032t^{12} - 260469752264271888t^{11} - 24373891414466292t^{10} + 4700991886227076t^{9} + 1132868822547275t^{8} + 170608810656800t^{7} + 5838252288027t^{6} + 819861084455t^{5} + 275658140877t^{4} + 7894311675t^{3} + 1512873954t^{2} + 7307658t + 124488 \right)D^{5} + \left( 9947792272218402201600t^{22} + 17022844506448168558080t^{21} + 16405461511450521207808t^{20} + 13256042852632210888448t^{19} + 8049408752970848817408t^{18} + 3390196637045604118704t^{17} + 1021487224765415539424t^{16} + 105521552558617529376t^{15} - 47497768736765040728t^{14} - 27602237486636736492t^{13} - 7648191433695909224t^{12} - 903212181871569576t^{11} - 34839312909078580t^{10} + 30053347210172078t^{9} + 6653778238378777t^{8} + 1058472366813427t^{7} + 97402686869705t^{6} + 7927281149728t^{5} - 296496270868t^{4} - 4076492316t^{3} - 1442969481t^{2} - 7568100t - 62244 \right)D^{4} + t \left( 41780727543317289246720t^{21} + 69176723007592405693440t^{20} + 61826999763983470737920t^{19} + 48386180759340910513152t^{18} + 28284088577945923739520t^{17} + 10060621213590935584608t^{16} + 2170280697239815137824t^{15} - 173867617620849549440t^{14} - 281020440646053307544t^{13} - 89525292922375453880t^{12} - 18093556748349134280t^{11} - 1138022952800188472t^{10} + 156996727874379092t^{9} + 110887122650032816t^{8} + 15908935128135493t^{7} + 1863956537102524t^{6} + 153048610900589t^{5} + 9032439889947t^{4} + 115673074875t^{3} + 2410518903t^{2} + 9116430t + 19836 \right)D^{3} + 2 t ^{2} \left( 46157756143093386215424t^{20} + 74463511866914192721024t^{19} + 62898332433171751821056t^{18} + 48436002914752832866496t^{17} + 27597750173340562803008t^{16} + 8189688588293342936704t^{15} + 998646891995467090496t^{14} - 484169839842064608736t^{13} - 305478820719696780168t^{12} - 71925610178819153056t^{11} - 11541972364311591168t^{10} - 99443414769201040t^{9} + 213662045550904026t^{8} + 88490321758804656t^{7} + 10737825265782493t^{6} + 1244911260926850t^{5} + 97200203815692t^{4} + 4561516903717t^{3} + 71140660348t^{2} + 1631539911t + 3017469 \right)D^{2} + 4 t ^{2} \left( 25068436525990373548032t^{20} + 39661196595870293898624t^{19} + 32167479129849448519424t^{18} + 24611214038768581567808t^{17} + 13775764447700517271872t^{16} + 3415209171870130512448t^{15} + 77844684002099873536t^{14} - 328482206199306583872t^{13} - 151283508744969539624t^{12} - 28988714401928280144t^{11} - 3906635414786978280t^{10} + 211232628838923592t^{9} + 111907642867471396t^{8} + 36050836767299184t^{7} + 3938396765273544t^{6} + 474925020235488t^{5} + 32886524899116t^{4} + 1235721070854t^{3} + 24045572727t^{2} + 497513556t + 373464 \right)D + 96 t ^{3} \left( 426333954523645808640t^{19} + 665545922628089791680t^{18} + 525280236562460505472t^{17} + 401165863846119419104t^{16} + 221882863622764323744t^{15} + 47026254208199567648t^{14} - 3410179156571307304t^{13} - 6028475095560771072t^{12} - 2344249887463229692t^{11} - 387518993960601720t^{10} - 46105231949291658t^{9} + 5580060319506392t^{8} + 1774796856765392t^{7} + 491994206881152t^{6} + 50581689319380t^{5} + 6377495071848t^{4} + 382164951384t^{3} + 11859622902t^{2} + 281545065t + 5041764 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 5)

Sector dimension: 9
Spectral radius: $8.6538925$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$8.653892$	1	1
$-3.155362 - 6.590098\mathrm{i}$	1	1
$-3.155362 + 6.590098\mathrm{i}$	1	1
$-1.96694 - 2.805953\mathrm{i}$	1	1
$-1.96694 + 2.805953\mathrm{i}$	1	1
$0 - 2\mathrm{i}$	1	1
$0 + 2\mathrm{i}$	1	1
$-0.7046445 - 1.721586\mathrm{i}$	1	1
$-0.7046445 + 1.721586\mathrm{i}$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda^{2} + 4\right) \left(\lambda^{7} + 3 \lambda^{6} + 7 \lambda^{5} - 487 \lambda^{4} - 2524 \lambda^{3} - 9717 \lambda^{2} - 13984 \lambda - 18772\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 12 & 12 & 9 & 8 & 8 & 10 \\ 1 & -1 & 0 & 0 & 0 & 4 & 6 & 6 & 0 \\ 2 & 2 & 0 & 0 & 0 & -2 & 9 & 3 & 8 \\ 0 & 2 & \frac{5}{2} & \frac{1}{2} & \frac{3}{4} & 0 & -2 & 0 & \frac{3}{2} \\ 0 & 1 & -1 & -1 & -\frac{3}{2} & 0 & 4 & 0 & 3 \\ 0 & 0 & 2 & 2 & 3 & 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & 2 & 0 & \frac{5}{2} & \frac{1}{2} & \frac{3}{2} & 0 \\ 0 & 0 & 0 & 4 & 5 & -\frac{1}{2} & -\frac{1}{2} & -\frac{3}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{2} & \frac{5}{2} & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta