Family 2-351-75-a

← 2-352-75-d · 2-325-70-a →

Blow-up of $\mathbb P^4$ along a smooth cubic scroll.

$\pi_1\colon X\to\mathbb P^4$ is the blow-up of $\mathbb P^4$ (1-625-126-a) along a smooth rational normal cubic scroll $S(1,2)\cong\mathbb F_1$.
$\pi_2\colon X\to\mathbb P^2$ is a $\mathbb P^2$-bundle, the projectivization of the rank-$3$ kernel of a general surjection $\mathcal O^{\oplus5}\to\mathcal O(1)^{\oplus2}$ on $\mathbb P^2$.

More details (pdf)

Varieties in this family can be blown up to

Varieties in this family can be blown down to

1-625-126-a

Defining data

Ambient: ℙ²P2 × ℙ⁴P4
Bundle: $(E^{ω_{1} + ω_{3}})^{\oplus 2}$ 2 copies of E^(w1 + w3)
Bundle: $(O_{1} (1) ⊠ O_{2} (1))^{\oplus 2}$ 2 copies of O_1(1) box O_2(1)
ZL62 label: 2040.hh

Line bundles on the ambient $Y$

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

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	75
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	351
$\chi(X, \mathrm{T}_X)$	=	6

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	351
$\mathrm{c}_1^2\mathrm{c}_2$	=	186
$\mathrm{c}_2^2$	=	94
$\mathrm{c}_1\mathrm{c}_3$	=	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 - 6	0	x₀ - 6	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 61	0	0	x₁ - 39	100
$\mathrm{HH}^{ 4 }$ = 75	0	0	0	0	75
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 70t + 209t^{2} + 70t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(13)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(18)

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( 30735t^{9} + 28118t^{8} + 29945t^{7} + 21786t^{6} + 11002t^{5} + 2159t^{4} + 77t^{3} - 6t^{2} - 4t - 1 \right) \left( 1730784544985521t^{13} + 6274843973419832t^{12} + 12024818181069478t^{11} + 14382117325118704t^{10} + 10669713395791916t^{9} + 5019186889336813t^{8} + 1552919435492392t^{7} + 327079140220366t^{6} + 48370857640910t^{5} + 5092976598923t^{4} + 369922786965t^{3} + 14736976965t^{2} + 58429350t + 3316950 \right)D^{6} + \left( 1117108922792729746635t^{22} + 5070186646717636754566t^{21} + 12350021689086746939243t^{20} + 20002952626241768098150t^{19} + 23149764201796148728320t^{18} + 20501524856495822887369t^{17} + 14537829020755843914373t^{16} + 8355866872872117411010t^{15} + 3858324280894586121076t^{14} + 1410092710599204175645t^{13} + 403148922357506780565t^{12} + 89798391793626566392t^{11} + 15721534311256970105t^{10} + 2236006259159848432t^{9} + 277346495353884390t^{8} + 32681061258511615t^{7} + 3729679738297427t^{6} + 384055995999474t^{5} + 31824612964488t^{4} + 1850415105825t^{3} + 58594780260t^{2} + 148752450t + 6633900 \right)D^{5} + \left( 9309241023272747888625t^{22} + 42239505590509953078770t^{21} + 102204312609404810120279t^{20} + 162175671086501341650166t^{19} + 179083622716782701267094t^{18} + 146949204208885952402704t^{17} + 94609609191673160009919t^{16} + 49066411740247697816633t^{15} + 20575660635215286006302t^{14} + 6933275548270161752198t^{13} + 1862556643116417601358t^{12} + 395660972628762645597t^{11} + 65607759970741513373t^{10} + 8217920081252219609t^{9} + 706211535123516872t^{8} + 26699981007590744t^{7} - 2454042674807110t^{6} - 467741570809631t^{5} - 35447546633057t^{4} - 1734183602595t^{3} - 55601581635t^{2} - 104526450t - 3316950 \right)D^{4} + t \left( 39098812297745541132225t^{21} + 177366159144089637340210t^{20} + 428538233797192784249845t^{19} + 673907898860862375832290t^{18} + 722665247974019834742988t^{17} + 560918640232380204850009t^{16} + 334228428152386118406923t^{15} + 158693729957503377217924t^{14} + 60963585051865935652736t^{13} + 18998496696078750525277t^{12} + 4796123076595382807889t^{11} + 977550509181570821820t^{10} + 159747721738077000647t^{9} + 20587466761910595652t^{8} + 2020516888458893022t^{7} + 140219522584387455t^{6} + 5891323774529957t^{5} + 112854209639726t^{4} + 1556735660790t^{3} + 23421228615t^{2} + 4674600t + 935550 \right)D^{3} + 2 t ^{2} \left( 43194878347985550203220t^{20} + 195913765484664459383776t^{19} + 473835767260595047849907t^{18} + 743294505265365991158250t^{17} + 783773758022828894871575t^{16} + 585586737691896591201768t^{15} + 329184952137993611950095t^{14} + 145610943881429770830519t^{13} + 51945168780318100019600t^{12} + 15096896110719443148226t^{11} + 3588724490415318087487t^{10} + 697569676404043828786t^{9} + 110067560455755987662t^{8} + 13802857794084197446t^{7} + 1318555307400061270t^{6} + 88563101542708342t^{5} + 3573451402444613t^{4} + 69929007117926t^{3} + 1560790596560t^{2} + 22709549565t - 1228500 \right)D^{2} + 4 t ^{3} \left( 23459287378647324679335t^{19} + 106388064764594105229766t^{18} + 257781339227871194043948t^{17} + 404579464406168312936870t^{16} + 422807870567474852388323t^{15} + 308158339642050182302545t^{14} + 166242919425876138565316t^{13} + 69734301766111929670937t^{12} + 23472551047338667532183t^{11} + 6441603312684069989807t^{10} + 1452825323829265908025t^{9} + 269883461660030704198t^{8} + 41015602725449956735t^{7} + 4979530471619843055t^{6} + 460299853562874057t^{5} + 29781868432105161t^{4} + 1147265671840344t^{3} + 21875006142387t^{2} + 591559049340t + 7039231290 \right)D + 24 t ^{3} \left( 1595869889703899638050t^{19} + 7236668531666077572980t^{18} + 17564163253097084203164t^{17} + 27602428008585838555486t^{16} + 28715299860179440378180t^{15} + 20612964606861222088716t^{14} + 10824677634671878701295t^{13} + 4378314590874600969952t^{12} + 1413797026159495408447t^{11} + 371769695285184777706t^{10} + 80453842037687987528t^{9} + 14380930899300755837t^{8} + 2110461109474091111t^{7} + 248065636439323728t^{6} + 22186344644724366t^{5} + 1386592373333385t^{4} + 51509180584890t^{3} + 968300475660t^{2} + 29067198300t + 268672950 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ²P2 × ℙ⁴P4

Sector dimension: 9
Spectral radius: $7.30381074$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$7.303811$	1	1
$-0.07855329 - 6.574122\mathrm{i}$	1	1
$-0.07855329 + 6.574122\mathrm{i}$	1	1
$-4.451389 - 0.01907406\mathrm{i}$	1	1
$-4.451389 + 0.01907406\mathrm{i}$	1	1
$-1.298435 - 1.209079\mathrm{i}$	1	1
$-1.298435 + 1.209079\mathrm{i}$	1	1
$0.1764722 - 1.236782\mathrm{i}$	1	1
$0.1764722 + 1.236782\mathrm{i}$	1	1

Characteristic polynomial

$$\begin{equation}\lambda^{9} + 4 \lambda^{8} + 6 \lambda^{7} - 77 \lambda^{6} - 2159 \lambda^{5} - 11002 \lambda^{4} - 21786 \lambda^{3} - 29945 \lambda^{2} - 28118 \lambda - 30735\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

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

Fano fourfolds of type A beta