Family 2-244-55-a

← 2-256-57-b · 2-240-54-a →

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

$\pi_1\colon X\to Y$ is the blow-up $\mathrm{Bl}_{\ell}Y$ of a smooth intersection $Y\subset\mathbb P^6$ of two quadrics (1-324-70-a) along a line $\ell\subset Y$.
$\pi_2\colon X\to\mathbb P^4$ is the blow-up $\mathrm{Bl}_S\mathbb P^4$ of $\mathbb P^4$ (1-625-126-a) along a smooth quintic Castelnuovo surface $S$, a rational surface of sectional genus $2$, isomorphic to the blow-up of $\mathbb P^2$ at eight points and embedded by $|4l-2e_0-e_1-\cdots-e_7|$; it resolves the projection of $Y$ (1-324-70-a) from $\ell$.

More details (pdf)

Varieties in this family can be blown down to

ℙ⁴P4 × ℙ⁶P6 · 40600.wOw3w3 (preferred model)

Defining data

Ambient: ℙ⁴P4 × ℙ⁶P6
Bundle: $(E^{ω_{1} + ω_{5}})^{\oplus 2} \oplus E^{ω_{4} + ω_{5}}$ 2 copies of E^(w1 + w5) + E^(w4 + w5)
Bundle: $(O_{1} (1) ⊠ O_{2} (1))^{\oplus 2} \oplus Q_{1} ⊠ O_{2} (1)$ 2 copies of O_1(1) box O_2(1) + Q_1 box O_2(1)
ZL62 label: 40600.wOw3w3

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;7) \\ \det\mathcal{E} &\cong \mathcal{O}(3;6) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	55
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	244
$\chi(X, \mathrm{T}_X)$	=	-8

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	244
$\mathrm{c}_1^2\mathrm{c}_2$	=	160
$\mathrm{c}_2^2$	=	100
$\mathrm{c}_1\mathrm{c}_3$	=	40
$\mathrm{c}_4$	=	16

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		10		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 + 8	0	x₀ + 8	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 48	0	0	x₁ - 12	60
$\mathrm{HH}^{ 4 }$ = 55	0	0	0	0	55
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 50t + 142t^{2} + 50t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(31)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(36)

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( 10037t^{7} + 12975t^{6} + 13475t^{5} + 4109t^{4} + 492t^{3} - 9t^{2} - 2t - 1 \right) \left( 369138016637646t^{13} + 879915907782336t^{12} + 1463082784937226t^{11} + 1564013797325697t^{10} + 823030819101309t^{9} + 200777689749551t^{8} + 44239058305653t^{7} + 23792040974246t^{6} + 7566032210524t^{5} + 1085286521560t^{4} + 64360215660t^{3} + 3895285404t^{2} + 20160080t + 463600 \right)D^{6} + \left( t + 1 \right) \left( 77805803732833110942t^{21} + 323885866829794195680t^{20} + 774672351735227171874t^{19} + 1276400059755200167392t^{18} + 1477814604241015799463t^{17} + 1211217571707323356532t^{16} + 700872095549661718437t^{15} + 288436267219241591680t^{14} + 93522610181348610831t^{13} + 31418927815128116819t^{12} + 11616495032688749592t^{11} + 3488387860303064084t^{10} + 708481100868046388t^{9} + 92690377029751761t^{8} + 8107343115071069t^{7} + 640277754332328t^{6} + 68805046796520t^{5} + 6362488872772t^{4} + 307172350044t^{3} + 15405714656t^{2} + 55844240t + 927200 \right)D^{5} + \left( 648381697773609257850t^{22} + 3080984899481767927290t^{21} + 7946443391333884249614t^{20} + 14108733162276083248977t^{19} + 17739896619323074395450t^{18} + 15776174489846717985534t^{17} + 9958751612078550441872t^{16} + 4542987665688740848212t^{15} + 1624545058018195184151t^{14} + 565074653511075796751t^{13} + 219823476307069158573t^{12} + 77447864395769244838t^{11} + 19760186281519672227t^{10} + 3365341459900459917t^{9} + 379841633876000433t^{8} + 29965573476565969t^{7} + 1451783773682786t^{6} - 21296966900164t^{5} - 7912978086120t^{4} - 288453512956t^{3} - 14422049076t^{2} - 47587280t - 463600 \right)D^{4} + t \left( 2723203130649158882970t^{21} + 12060865805641008522570t^{20} + 29766412271288065802430t^{19} + 50862804662882613739314t^{18} + 60298705525243366874853t^{17} + 49093522778548813570819t^{16} + 27687931674775476681889t^{15} + 11308616330137597751267t^{14} + 3924897272548294048361t^{13} + 1520037827202620542240t^{12} + 635078100833081731869t^{11} + 214500850703128033832t^{10} + 50392731770010100420t^{9} + 7877572517392765527t^{8} + 824383275049950624t^{7} + 63556221263960987t^{6} + 3564260871765764t^{5} + 105098754444148t^{4} + 1045512145336t^{3} + 15301975444t^{2} + 25481120t + 220400 \right)D^{3} + 2 t ^{2} \left( 3008491077669546956424t^{20} + 12581191927839690477132t^{19} + 30069261354806092582206t^{18} + 49891325725741719158820t^{17} + 56163976547055768824979t^{16} + 42218835033125998782151t^{15} + 21518635769170272208384t^{14} + 8051487365737314735312t^{13} + 2848088268133666227913t^{12} + 1229508844777099662928t^{11} + 521912633830664031324t^{10} + 165326394471068269452t^{9} + 35800097174124919508t^{8} + 5180466208471847371t^{7} + 511833036818606768t^{6} + 38664648426637100t^{5} + 2040063533620192t^{4} + 54433210525452t^{3} + 757762016496t^{2} + 12791339236t + 10077200 \right)D^{2} + 8 t ^{2} \left( 816960939194747664891t^{20} + 3268549775483386795161t^{19} + 7637439448244288302722t^{18} + 12394658370326943228606t^{17} + 13362952040403809741733t^{16} + 9382473702396092048334t^{15} + 4386079101517727751618t^{14} + 1540979400333841604844t^{13} + 574535403576547954866t^{12} + 269186123608225299741t^{11} + 112394348922781799720t^{10} + 33378234870796066560t^{9} + 6732557001235939756t^{8} + 913794981036642876t^{7} + 86790911459653416t^{6} + 6445280439838168t^{5} + 311113484088816t^{4} + 7348973804616t^{3} + 127469658560t^{2} + 1957829312t + 741760 \right)D + 288 t ^{3} \left( 9262595682480132255t^{19} + 35925540161914279305t^{18} + 82699640165047498512t^{17} + 132138649200453350454t^{16} + 137879980562588032289t^{15} + 91792021515516291000t^{14} + 40065394861348611736t^{13} + 13504906564441612044t^{12} + 5359896702775000050t^{11} + 2644104907112087545t^{10} + 1074747864867121944t^{9} + 302237852478497724t^{8} + 57573268159355876t^{7} + 7426326237116306t^{6} + 687601777149280t^{5} + 50258880400788t^{4} + 2205331954848t^{3} + 46460775312t^{2} + 953532240t + 12517200 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ⁴P4 × ℙ⁶P6

Show multiplicities

Sector dimension: 9
Spectral radius: $9.41672697$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$9.416727$	1	1
$-1.87253 - 7.82709\mathrm{i}$	1	1
$-1.87253 + 7.82709\mathrm{i}$	1	1
$-3.353089 - 2.055696\mathrm{i}$	1	1
$-3.353089 + 2.055696\mathrm{i}$	1	1
$-0.4827438 - 0.9114692\mathrm{i}$	1	1
$-0.4827438 + 0.9114692\mathrm{i}$	1	1
$-1$	2	2

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda^{7} + 2 \lambda^{6} + 9 \lambda^{5} - 492 \lambda^{4} - 4109 \lambda^{3} - 13475 \lambda^{2} - 12975 \lambda - 10037\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 12 & 12 & 12 & 48 & 40 & 20 \\ 2 & 0 & 3 & 0 & -2 & 0 & 0 & 3 & 32 \\ 1 & 0 & -1 & 0 & 2 & 0 & 24 & 21 & 16 \\ 0 & 1 & 2 & -1 & 0 & 0 & 0 & -2 & 0 \\ 0 & 2 & 0 & 3 & 0 & 4 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & -1 & 0 & 2 & 12 \\ 0 & 0 & 0 & 1 & -\frac{5}{2} & 3 & -6 & -\frac{15}{4} & 0 \\ 0 & 0 & 0 & 2 & 5 & 0 & 8 & 5 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & \frac{5}{2} & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta