Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-101-27-a

·

This is a Fano fourfold of K3 type, appearing as case C-9 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ5P5 × Gr(2, 4)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7 (marked)node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
2 copies of E^(w1 + w6) + E^(w1 + w7)
Bundle
2 copies of O_1(1) box U_2^v + O_1(1) box O_2(1)
ZL62 label
3150.w8w2w2
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(6;4) \\ \det\mathcal{E} &\cong \mathcal{O}(5;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 27
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 101
$\chi(X, \mathrm{T}_X)$ = -21

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 101
$\mathrm{c}_1^2\mathrm{c}_2$ = 110
$\mathrm{c}_2^2$ = 130
$\mathrm{c}_1\mathrm{c}_3$ = 22
$\mathrm{c}_4$ = 31
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 23 1 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 + 21 0 x0 + 21 x1
$\mathrm{HH}^{ 3 }$ = x1 + 19 0 0 x1 + 10 9
$\mathrm{HH}^{ 4 }$ = 28 0 0 1 0 27
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 22t + 55t^{2} + 22t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(579)
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) ^{3} \left( 5t^{2} + 2t + 1 \right) \left( 3913t^{5} + 5886t^{4} + 2960t^{3} + 309t^{2} - 3t - 1 \right) \left( 295415991460820t^{14} + 1320788173401844t^{13} + 2568038778692367t^{12} + 2917261668146220t^{11} + 2179121207783862t^{10} + 1123855913160552t^{9} + 400764833546789t^{8} + 96598106493032t^{7} + 15455640350672t^{6} + 1703354703892t^{5} + 152874490093t^{4} + 11643563660t^{3} + 390720622t^{2} + 4124720t + 14039 \right)D^{6} + \left( t + 1 \right) ^{2} \left( 121376091331549809300t^{22} + 824966928976220301160t^{21} + 2582394573489022250957t^{20} + 4969147751143836944774t^{19} + 6609686905151316003535t^{18} + 6465814694276808652145t^{17} + 4818697191073127522722t^{16} + 2789406066233893858076t^{15} + 1265464795647269114892t^{14} + 450881252132069513760t^{13} + 125972763069196161658t^{12} + 27575408931909725096t^{11} + 4764108052214545814t^{10} + 666828011557290150t^{9} + 79067045231099760t^{8} + 8069218647273476t^{7} + 653227497645984t^{6} + 36580408210808t^{5} + 1530097269561t^{4} + 62766775890t^{3} + 1541840155t^{2} + 12135497t + 28078 \right)D^{5} + \left( t + 1 \right) \left( 1011467427762915077500t^{23} + 7418183989529893243700t^{22} + 25055631664303329723463t^{21} + 51990468540807329652328t^{20} + 74528853161626365255904t^{19} + 78566207813682086118801t^{18} + 63147964767349055232929t^{17} + 39489309230274976693441t^{16} + 19393961837957617488312t^{15} + 7494970152966362748320t^{14} + 2273801960316572294746t^{13} + 539835186354228289076t^{12} + 100471283220883020972t^{11} + 14878492039127260234t^{10} + 1811202295415994986t^{9} + 186482241424404726t^{8} + 15738514246715996t^{7} + 961054367811404t^{6} + 34168719018447t^{5} + 431155869892t^{4} - 35587919964t^{3} - 1294819387t^{2} - 9421627t - 14039 \right)D^{4} + t \left( 4248163196604243325500t^{23} + 33860110059614764615600t^{22} + 124465619642229279579775t^{21} + 281324212093583109613404t^{20} + 439619461797529379928724t^{19} + 505743942351501256097737t^{18} + 444455337793670046343749t^{17} + 304844498369182988198953t^{16} + 164944023620975756199784t^{15} + 70628098581528484570204t^{14} + 23894966965020813972050t^{13} + 6366546438353594077540t^{12} + 1335238625966154262440t^{11} + 222551706006091251802t^{10} + 30316357139591454582t^{9} + 3510206110062996346t^{8} + 346999666614506844t^{7} + 27244256341716532t^{6} + 1513127516948335t^{5} + 53602344043552t^{4} + 1191475619396t^{3} + 15284008733t^{2} + 70811269t + 20989 \right)D^{3} + 2 t ^{2} \left( 4693208864819925959600t^{22} + 36101952084859321460160t^{21} + 127822016232376686086223t^{20} + 277930066580059845723430t^{19} + 417495902810615153620865t^{18} + 461375839918652704431099t^{17} + 389006902461678449643676t^{16} + 255359119654396654273426t^{15} + 131716086475477993323512t^{14} + 53485906718324401436148t^{13} + 17063478051762718621966t^{12} + 4267576879432511545558t^{11} + 839331208128587565434t^{10} + 131944582001795813126t^{9} + 17152655635986251124t^{8} + 1901862638251896558t^{7} + 175255480840467688t^{6} + 12181793040039868t^{5} + 587288282686275t^{4} + 20304991089428t^{3} + 461799829221t^{2} + 6395615743t + 29459680 \right)D^{2} + 4 t ^{2} \left( 2548897917962545995300t^{22} + 19087545439052357367760t^{21} + 65676110193833957140522t^{20} + 138627854515543631528024t^{19} + 202018702677008625870637t^{18} + 216431930763695049402448t^{17} + 176680406638402447724018t^{16} + 112019270248316426139703t^{15} + 55600031710561159150256t^{14} + 21624589885298314386859t^{13} + 6576625464176028986412t^{12} + 1562942080691319005143t^{11} + 292300580130478055664t^{10} + 44006372476232505797t^{9} + 5533904463793702394t^{8} + 590981955738533761t^{7} + 50725420209931544t^{6} + 3128779829148861t^{5} + 132679179165606t^{4} + 4650475621969t^{3} + 104555021367t^{2} + 1358886331t + 5952536 \right)D + 64 t ^{2} \left( 65022906070473112125t^{22} + 477972947813168727300t^{21} + 1612024671005754067398t^{20} + 3332025773836835534817t^{19} + 4751752085555269119652t^{18} + 4978185320161372955358t^{17} + 3968886549537321029359t^{16} + 2452038511963918255158t^{15} + 1182035860530326934830t^{14} + 444717089381622559093t^{13} + 130318395215470277369t^{12} + 29764809045480652089t^{11} + 5355050657079437400t^{10} + 780477961492715990t^{9} + 95692972503002997t^{8} + 9884452710560058t^{7} + 795077173940569t^{6} + 44335426019451t^{5} + 1720545106889t^{4} + 62162033086t^{3} + 1350843832t^{2} + 14782928t + 56156 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ5P5 × Gr(2, 4)

|λ| = 20.2344 λ ≈ 20.2344; multiplicity 1; Jordan blocks 1 λ ≈ -10.40893 - 0.8412631i; multiplicity 1; Jordan blocks 1 λ ≈ -10.40893 + 0.8412631i; multiplicity 1; Jordan blocks 1 λ ≈ -1 - 2i; multiplicity 1; Jordan blocks 1 λ ≈ -1 + 2i; multiplicity 1; Jordan blocks 1 λ ≈ -1.208268 - 0.5598016i; multiplicity 1; Jordan blocks 1 λ ≈ -1.208268 + 0.5598016i; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 3; Jordan blocks 1, 2 3
Sector dimension
10
Spectral radius
$20.2343955$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$20.2344$11
$-10.40893 - 0.8412631\mathrm{i}$11
$-10.40893 + 0.8412631\mathrm{i}$11
$-1 - 2\mathrm{i}$11
$-1 + 2\mathrm{i}$11
$-1.208268 - 0.5598016\mathrm{i}$11
$-1.208268 + 0.5598016\mathrm{i}$11
$-1$31, 2

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{3} \left(\lambda^{2} + 2 \lambda + 5\right) \left(\lambda^{5} + 3 \lambda^{4} - 309 \lambda^{3} - 2960 \lambda^{2} - 5886 \lambda - 3913\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda^{2} + 2 \lambda + 5\right) \left(\lambda^{5} + 3 \lambda^{4} - 309 \lambda^{3} - 2960 \lambda^{2} - 5886 \lambda - 3913\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 16 & 96 & 90 & 48 & 30 & 412 & 300 & 675 \\ 1 & -1 & 2 & 28 & 32 & 10 & 6 & 84 & 54 & 168 \\ 1 & 3 & -1 & 22 & 12 & 16 & 10 & 51 & 45 & 132 \\ 0 & 1 & 1 & -2 & 0 & -1 & 0 & 16 & 12 & 9 \\ 0 & 1 & 0 & 2 & -1 & 3 & 1 & 6 & 6 & 18 \\ 0 & 0 & 1 & 3 & 5 & 0 & 0 & 20 & 12 & 27 \\ 0 & 0 & 1 & 3 & 3 & -1 & -1 & -6 & -6 & 9 \\ 0 & 0 & 0 & 0 & -\frac{5}{2} & 2 & 1 & -\frac{3}{2} & \frac{3}{2} & 0 \\ 0 & 0 & 0 & \frac{14}{3} & \frac{43}{6} & -\frac{2}{3} & \frac{1}{3} & \frac{23}{6} & -\frac{1}{2} & 16 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{8}{3} & 2 & 0\end{pmatrix}\notag\end{equation}$$