Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-110-29-f

·

This is a Fano fourfold of K3 type, appearing as case K3-52 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ2P2 × ℙ2P2 × ℙ3P3
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(w3 + w5) + E^(w1 + w5) + E^(w1 + w3 + w5)
Bundle
O_2(1) box O_3(1) + O_1(1) box O_3(1) + O_1(1) box O_2(1) box O_3(1)
ZL62 label
202030.wCw3x4600
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 29
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 110
$\chi(X, \mathrm{T}_X)$ = -20

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 110
$\mathrm{c}_1^2\mathrm{c}_2$ = 116
$\mathrm{c}_2^2$ = 124
$\mathrm{c}_1\mathrm{c}_3$ = 28
$\mathrm{c}_4$ = 34
Hodge diamond
1
00
0 3 0
0 0 0 0
0 1 24 1 0
0 0 0 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 + 20 0 x0 + 20 x1
$\mathrm{HH}^{ 3 }$ = x1 + 17 0 0 x1 + 9 8
$\mathrm{HH}^{ 4 }$ = 30 0 0 1 0 29
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 24t + 60t^{2} + 24t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(560)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(590)

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( 3t + 1 \right) ^{3} \left( 51t^{3} + 26t^{2} + 5t + 1 \right) \left( 2579t^{4} + 1418t^{3} + 184t^{2} + 7t - 1 \right) \left( 5231378381523258t^{14} + 7254834017739254t^{13} - 1959886657897899t^{12} - 10954545745291120t^{11} - 10437425130221611t^{10} - 5233930510803380t^{9} - 1580588095525503t^{8} - 290510642734934t^{7} - 30358153568774t^{6} - 1505137913616t^{5} - 42675090221t^{4} - 6995315624t^{3} - 513309020t^{2} + 2395936t - 24420 \right)D^{6} + \left( 3t + 1 \right) ^{2} \left( 43348911930032473893366t^{22} + 108420016855262044828716t^{21} + 67118694235473654405633t^{20} - 92467289482710630369198t^{19} - 218565252202346638474270t^{18} - 216473645074094497028026t^{17} - 136216842336487201231871t^{16} - 60458384481115928147537t^{15} - 19753261308423166350118t^{14} - 4836343000858837384980t^{13} - 889695267865375866288t^{12} - 121864052463250682410t^{11} - 12271229813542213183t^{10} - 936453697781673501t^{9} - 68261383662519015t^{8} - 6444992568947159t^{7} - 563873469917908t^{6} - 27640031963171t^{5} - 517947764698t^{4} - 25636125920t^{3} - 2121461136t^{2} + 7602948t - 48840 \right)D^{5} + \left( 3t + 1 \right) \left( 1083722798250811847334150t^{23} + 2830154743587649965236910t^{22} + 1858643213710627282284441t^{21} - 2557252563507480259302624t^{20} - 6424969346581838230108989t^{19} - 6699407293113042552279092t^{18} - 4435437022363540754880439t^{17} - 2070805853248536972534467t^{16} - 711238110474225639698011t^{15} - 182885457343435591846459t^{14} - 35322414320879553991688t^{13} - 5088495796454341059014t^{12} - 540135874182953217973t^{11} - 42454660637126152581t^{10} - 2754197393226440508t^{9} - 196575160873929040t^{8} - 15487615519757589t^{7} - 879430952934814t^{6} - 23973271621354t^{5} + 40742498341t^{4} + 34168546636t^{3} + 2154773572t^{2} - 1778888t + 24420 \right)D^{4} + t \left( 13654907257960229276410290t^{23} + 37819878077470498785324000t^{22} + 27812897771671991654005545t^{21} - 33195913242776264336698008t^{20} - 94004220104584425729817455t^{19} - 105263068477203772578390276t^{18} - 74574446779065360207799447t^{17} - 37322915288346580622985579t^{16} - 13792468993777755988838685t^{15} - 3835806096409389012150581t^{14} - 807125894401346008656234t^{13} - 128087714980286436084656t^{12} - 15258403119913089854941t^{11} - 1388963775610852828963t^{10} - 108248336662813154922t^{9} - 9208842373613140312t^{8} - 865672581652575033t^{7} - 65808143553502632t^{6} - 3116367617105802t^{5} - 69841589934433t^{4} - 271673584904t^{3} - 5739363804t^{2} - 189517296t - 58164 \right)D^{3} + 2 t ^{2} \left( 15085421351651300914891368t^{22} + 39760398298068618443572476t^{21} + 24679085621630650826058261t^{20} - 43167926604820294817575542t^{19} - 104585109450538703007459291t^{18} - 110427527720521243048730352t^{17} - 74556379659955090118476961t^{16} - 35547917111099108608226951t^{15} - 12459932531186175385182008t^{14} - 3263864975579770131641351t^{13} - 641184187702827090725115t^{12} - 94195791907016160394700t^{11} - 10397517792426093225199t^{10} - 913500928795638243978t^{9} - 77240645279932139557t^{8} - 7458652525752927715t^{7} - 681414692843982258t^{6} - 44352904073320259t^{5} - 1683134811225472t^{4} - 27417574908308t^{3} - 79873561086t^{2} - 6751103532t - 61467340 \right)D^{2} + 8 t ^{2} \left( 4096472177388068782923087t^{22} + 10394716144223473286264730t^{21} + 5537584458060999561007137t^{20} - 12890983994718588875110590t^{19} - 28391924933561080549635903t^{18} - 28745530112297388563920305t^{17} - 18721533453418059714132277t^{16} - 8601217328640182375673433t^{15} - 2893526538685621325431680t^{14} - 723178552819592546902310t^{13} - 134593763586992501064569t^{12} - 18633522288594369960993t^{11} - 1954351491222204360896t^{10} - 172139830569836853818t^{9} - 15961559787263688127t^{8} - 1626914317646219162t^{7} - 138893700367164884t^{6} - 7862413449190295t^{5} - 246175339240368t^{4} - 2994665049396t^{3} - 14773456500t^{2} - 1030356140t - 5030520 \right)D + 16 t ^{2} \left( 836014730079197710800630t^{22} + 2065904328032914294611900t^{21} + 972932374709781203389932t^{20} - 2781738953421557334798492t^{19} - 5780780718136564276768200t^{18} - 5689848810161028246337628t^{17} - 3613891552813124415058877t^{16} - 1617006546143979624836454t^{15} - 528000386119864674533773t^{14} - 127469693561956743215342t^{13} - 22787498730136958517666t^{12} - 3020149278830755246171t^{11} - 306702177821553520171t^{10} - 27555963268939322725t^{9} - 2746538186277203627t^{8} - 283813372245603763t^{7} - 22677256707016546t^{6} - 1146390649248309t^{5} - 30582250583316t^{4} - 289640600232t^{3} - 3083574324t^{2} - 115132308t - 341880 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2 × ℙ3P3

|λ| = 20.03621 λ ≈ 20.03621; multiplicity 1; Jordan blocks 1 λ ≈ -5.242218 - 4.791804i; multiplicity 1; Jordan blocks 1 λ ≈ -5.242218 + 4.791804i; multiplicity 1; Jordan blocks 1 λ ≈ -1.209294 - 4.277176i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1.209294 + 4.277176i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -3; multiplicity 4; Jordan blocks 1, 1, 2 λ ≈ -2.581412; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2.55177; multiplicity 1; Jordan blocks 1 2 2 4 2
Sector dimension
14
Spectral radius
$20.0362058$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$20.03621$11
$-5.242218 - 4.791804\mathrm{i}$11
$-5.242218 + 4.791804\mathrm{i}$11
$-1.209294 - 4.277176\mathrm{i}$21, 1
$-1.209294 + 4.277176\mathrm{i}$21, 1
$-3$41, 1, 2
$-2.581412$21, 1
$-2.55177$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right)^{4} \left(\lambda^{3} + 5 \lambda^{2} + 26 \lambda + 51\right)^{2} \left(\lambda^{4} - 7 \lambda^{3} - 184 \lambda^{2} - 1418 \lambda - 2579\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right)^{2} \left(\lambda^{3} + 5 \lambda^{2} + 26 \lambda + 51\right) \left(\lambda^{4} - 7 \lambda^{3} - 184 \lambda^{2} - 1418 \lambda - 2579\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 8 & 12 & 48 & 60 & 27 & 60 & 27 & 48 & 320 & 140 & 224 & 735 \\ 1 & -1 & 1 & 2 & 6 & 10 & 2 & 14 & 6 & 12 & 57 & 27 & 36 & 132 \\ 1 & 1 & -1 & 2 & 6 & 14 & 6 & 10 & 2 & 12 & 57 & 24 & 42 & 132 \\ 1 & 2 & 2 & -1 & 6 & 6 & 4 & 6 & 4 & 4 & 63 & 27 & 45 & 144 \\ 0 & 1 & 1 & 0 & -2 & 2 & 0 & 2 & 0 & 3 & 8 & 2 & 8 & 18 \\ 0 & 1 & 0 & 1 & 2 & -2 & 0 & 1 & 1 & 0 & 4 & 2 & 2 & 18 \\ 0 & 1 & 0 & 0 & 1 & 2 & -1 & 0 & 0 & 1 & 14 & 6 & 10 & 27 \\ 0 & 0 & 1 & 1 & 2 & 1 & 1 & -2 & 0 & 0 & 4 & 0 & 6 & 18 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 2 & -1 & 1 & 14 & 6 & 10 & 27 \\ 0 & 0 & 0 & 1 & 0 & 2 & 1 & 2 & 1 & -1 & 6 & 4 & 2 & 18 \\ 0 & 0 & 0 & 0 & -6 & -4 & -12 & 24 & 9 & 19 & 15 & 13 & 3 & 12 \\ 0 & 0 & 0 & 0 & 10 & 6 & 15 & -26 & -9 & -22 & -16 & -16 & 0 & 0 \\ 0 & 0 & 0 & 0 & 5 & 5 & 9 & -15 & -6 & -11 & -8 & -7 & -2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{3} & 1 & \frac{5}{3} & 0\end{pmatrix}\notag\end{equation}$$