Family 3-305-66-b

← 3-305-66-a · 3-304-66-a →

Blow-up of family 2-352-75-c along a Hirzebruch surface $\mathbb F_1$.

$\pi_1\colon X\to Y_1$ is the blow-up $\mathrm{Bl}_{S_1}Y_1$ of $Y_1\cong\mathrm{Bl}_{\Pi_\Phi}W_5$, a Fano fourfold of type 2-352-75-c, where $W_5$ is the del Pezzo fourfold of degree $5$ (1-405-85-a) and $\Pi_K,\Pi_\Phi\subset W_5$ are two planes meeting transversally in one point; the centre $S_1\cong\mathbb F_1$ is the strict transform of $\Pi_K$.
$\pi_2\colon X\to Z\subset Q_U^3\times Q_P^3$, with $Q_U^3\subset\mathrm{Gr}(2,5)$ and $Q_P^3\subset\mathrm{Gr}(2,4)$ smooth quadric threefolds, maps birationally onto a determinantal fourfold; it contracts a unique plane $\Pi_0$ with normal bundle $\mathcal O_{\mathbb P^2}(-1)^{\oplus 2}$ to the unique singular point of $Z$.
$\pi_3\colon X\to Y_3$ is, symmetrically, the blow-up $\mathrm{Bl}_{S_3}Y_3$ of $Y_3\cong\mathrm{Bl}_{\Pi_K}W_5$, again of type 2-352-75-c, with $W_5$ (1-405-85-a) as above, along the strict transform $S_3\cong\mathbb F_1$ of $\Pi_\Phi$.

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Varieties in this family can be blown down to

2-352-75-c

Fl(2, 3, 5) × Gr(2, 4) · 3145.431w5 (preferred model)

Defining data

Ambient: Fl(2, 3, 5) × Gr(2, 4)
Bundle: $E^{ω_{6}} \oplus E^{ω_{1}} \oplus E^{ω_{2}} \oplus E^{ω_{4} + ω_{5}}$ E^(w6) + E^(w1) + E^(w2) + E^(w4 + w5)
Bundle: $O_{2} (1) \oplus U_{1, 3}^{\lor} \oplus O_{1} (1, 0) \oplus Q_{1, 1}^{\lor} \otimes U_{1, 2}^{\lor} \otimes ⋀^{2} U_{1, 3}^{\lor} ⊠ U_{2}^{\lor}$ O_2(1) + U_1,3^v + O_1(1,0) + Q_1,1^v ot U_1,2^v ot wedge^2 U_1,3^v box U_2^v
ZL62 label: 3145.431w5

Line bundles on the ambient $Y$

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

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	66
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	305
$\chi(X, \mathrm{T}_X)$	=	6

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	305
$\mathrm{c}_1^2\mathrm{c}_2$	=	170
$\mathrm{c}_2^2$	=	100
$\mathrm{c}_1\mathrm{c}_3$	=	58
$\mathrm{c}_4$	=	13

Hodge diamond

				1
			0		0
		0		3		0
	0		0		0		0
0		0		5		0		0
	0		0		0		0
		0		3		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 + 48	0	0	x₁ - 30	78
$\mathrm{HH}^{ 4 }$ = 66	0	0	0	0	66
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 61t + 181t^{2} + 61t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

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( 2t - 1 \right) \left( 13t^{2} + 5t + 1 \right) \left( 251t^{7} + 3465t^{6} - 2310t^{5} - 580t^{4} + 357t^{3} + 50t^{2} - t - 1 \right) \left( 2973699071779432t^{17} - 1886812294988368t^{16} - 1740182396385332t^{15} - 771013022227044t^{14} + 2397818981391140t^{13} - 1035688569107928t^{12} + 135632254486968t^{11} - 64883679368574t^{10} - 10604807901582t^{9} + 16873817698166t^{8} - 1465344011561t^{7} - 2119304719516t^{6} + 443770773957t^{5} - 16376284741t^{4} - 589742229t^{3} - 359369689t^{2} - 1027022t - 87535 \right)D^{6} + \left( 407533562991084037872t^{27} + 4245349602768435011432t^{26} - 6048081722827917858952t^{25} - 1696606974921992147484t^{24} + 1442373111937473516480t^{23} + 5409710383657493522108t^{22} - 4927067932540447232108t^{21} + 905239151741764196308t^{20} + 241624661168643325676t^{19} - 70575872683476115650t^{18} + 65418200203865477838t^{17} - 29192525007844470843t^{16} - 13297125793050850576t^{15} + 8183577372965410563t^{14} + 436933818097094905t^{13} - 770483204786906284t^{12} - 4783938282587140t^{11} + 37582620970539617t^{10} + 496778197104830t^{9} - 1347500813767045t^{8} + 59101025567420t^{7} + 18032032894862t^{6} - 1371648497191t^{5} + 169582039396t^{4} + 3638591485t^{3} + 1456479908t^{2} + 2380786t + 175070 \right)D^{5} + \left( 3396113024925700315600t^{27} + 28211521670141421389240t^{26} - 37944245996008497879768t^{25} - 13817505631817533236788t^{24} + 5505366815436643586964t^{23} + 39887661652844512816700t^{22} - 34898339582363128297268t^{21} + 8765483140736319880736t^{20} - 478216915653797227986t^{19} - 160705508131852781362t^{18} + 678238064973704815252t^{17} - 282869200503572412279t^{16} - 88775057006865067098t^{15} + 68248254006018622579t^{14} - 1983369190098787786t^{13} - 4776146618383644356t^{12} + 387583097340810309t^{11} + 143521543881963470t^{10} - 12669512995110464t^{9} - 2565983296737849t^{8} + 11711095971697t^{7} - 17689495994300t^{6} + 10167634156254t^{5} + 26706310208t^{4} + 2097154657t^{3} - 1410709087t^{2} - 7319346t - 87535 \right)D^{4} + t \left( 14263674704687941325520t^{26} + 94839298449928697604920t^{25} - 123401558313080023842200t^{24} - 55237754236964372190164t^{23} + 11049408641042183339472t^{22} + 144726218752956970200468t^{21} - 124888796380260318716468t^{20} + 36501755999284561296900t^{19} - 7865393368375408138532t^{18} + 1193907082972708593378t^{17} + 2697309184238917844310t^{16} - 1258373992335204798325t^{15} - 305450821383283075856t^{14} + 282503636088261081305t^{13} - 22022400931149132867t^{12} - 14088041542150832260t^{11} + 1852205809344801420t^{10} + 291046182690208329t^{9} - 41819761226596702t^{8} - 2998245339989693t^{7} - 243025628750828t^{6} + 24660716049868t^{5} + 7461096032813t^{4} + 136228422960t^{3} + 5017115091t^{2} + 88963584t + 13202 \right)D^{3} + 2 t ^{2} \left( 15757964435655249464384t^{25} + 84785730048358914397792t^{24} - 109118981071561978155488t^{23} - 56995966735788627049344t^{22} + 7026681284848786078920t^{21} + 136886193105986910774524t^{20} - 118116820717557164985688t^{19} + 37331060957070110875098t^{18} - 12244884737588719197784t^{17} + 2970093143041037110330t^{16} + 2595121423966819567094t^{15} - 1380321309715117052135t^{14} - 313523983915334389704t^{13} + 313451086490370820588t^{12} - 34193328904136201189t^{11} - 11087434011452094064t^{10} + 1862549268374256469t^{9} + 160404675717750536t^{8} - 25127919512733743t^{7} - 1184772287155183t^{6} - 397839465735108t^{5} + 11902437459400t^{4} + 3444243739722t^{3} + 51878632214t^{2} + 3078002579t + 35702690 \right)D^{2} + 4 t ^{2} \left( 8558204822812764795312t^{25} + 38091800893518933834632t^{24} - 49312832157724865916072t^{23} - 28786823474059492018540t^{22} + 3020625236931273380784t^{21} + 64095390562582281785164t^{20} - 55782349051471604260484t^{19} + 18377033377945720178054t^{18} - 7736466907027497551988t^{17} + 2275651947512568657660t^{16} + 1209065292275816675616t^{15} - 730129713985738775960t^{14} - 171385420525191525184t^{13} + 172591620246390351826t^{12} - 22156429788522615142t^{11} - 4403458447418374957t^{10} + 877205832341911948t^{9} + 48057656009150512t^{8} - 6017787940769127t^{7} - 501427545485985t^{6} - 216661929516351t^{5} + 3166133592072t^{4} + 835226541158t^{3} + 14582768258t^{2} + 976501021t + 7703080 \right)D + 8 t ^{2} \left( 1746572412818931590880t^{25} + 6676940230371894092880t^{24} - 8765886347981936859024t^{23} - 5523773022643622560872t^{22} + 608290716104174595888t^{21} + 11572123437395858101212t^{20} - 10190987645511055305720t^{19} + 3433203476897247695724t^{18} - 1704608647305607017132t^{17} + 556417198893938236260t^{16} + 216077916988014407540t^{15} - 145325182626810168798t^{14} - 36034742321278770900t^{13} + 35724253454768707431t^{12} - 4986153376396987974t^{11} - 693660874343275665t^{10} + 157553285928978109t^{9} + 6163869812135175t^{8} - 327808057043274t^{7} - 115838705524203t^{6} - 40902171146505t^{5} + 330936581790t^{4} + 82215803117t^{3} + 2155662108t^{2} + 128291583t + 700280 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 3, 5) × Gr(2, 4)

Show multiplicities

Sector dimension: 13
Spectral radius: $8.50650036$
Diagonalizable at 1: Yes
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$8.5065$	1	1
$-5.107352 - 4.360188\mathrm{i}$	1	1
$-5.107352 + 4.360188\mathrm{i}$	1	1
$-2.5 - 2.598076\mathrm{i}$	2	1, 1
$-2.5 + 2.598076\mathrm{i}$	2	1, 1
$-2.838993$	1	1
$2$	2	1, 1
$1.808246 - 0.2370073\mathrm{i}$	1	1
$1.808246 + 0.2370073\mathrm{i}$	1	1
$-0.06929548$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda - 2\right)^{2} \left(\lambda^{2} + 5 \lambda + 13\right)^{2} \left(\lambda^{7} + \lambda^{6} - 50 \lambda^{5} - 357 \lambda^{4} + 580 \lambda^{3} + 2310 \lambda^{2} - 3465 \lambda - 251\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda - 2\right) \left(\lambda^{2} + 5 \lambda + 13\right) \left(\lambda^{7} + \lambda^{6} - 50 \lambda^{5} - 357 \lambda^{4} + 580 \lambda^{3} + 2310 \lambda^{2} - 3465 \lambda - 251\right)\notag\end{equation}$$

Quantum multiplication matrix

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

Fano fourfolds of type A beta