Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-144-36-a

·

No geometric description is recorded for this family (yet).

Defining data
Ambient
ℙ3P3 × Gr(2, 7)
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_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9ω1ω2ω3ω4ω5ω6ω7ω8ω9
Bundle
E^(w3 + w4) + E^(w3 + w5)
Bundle
Q_1 box U_2^v + Q_1 box O_2(1)
ZL62 label
30601.wGwF
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;7) \\ \det\mathcal{E} &\cong \mathcal{O}(3;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 36
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 144
$\chi(X, \mathrm{T}_X)$ = -20

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 144
$\mathrm{c}_1^2\mathrm{c}_2$ = 132
$\mathrm{c}_2^2$ = 111
$\mathrm{c}_1\mathrm{c}_3$ = 26
$\mathrm{c}_4$ = 23
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 17 0 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 + x2 + 20 0 x0 + 20 x2
$\mathrm{HH}^{ 3 }$ = x1 + 24 0 0 x1 24
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 26 0 0 x1 - x2 - 10 0 36
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 31t + 80t^{2} + 31t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(255)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(323)

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( 108349t^{8} + 228295t^{7} + 190075t^{6} + 78640t^{5} + 16346t^{4} + 1453t^{3} + 32t^{2} - 1 \right) \left( 2869959281626290t^{13} + 8993231953841740t^{12} + 12606170810048875t^{11} + 10551605948863885t^{10} + 5931265984269105t^{9} + 2358219924034090t^{8} + 668843621489740t^{7} + 132480481023635t^{6} + 17719552196076t^{5} + 1572773792385t^{4} + 95844275547t^{3} + 3772320361t^{2} + 18046411t + 190800 \right)D^{6} + \left( 13060203164606929598820t^{22} + 70436921066784611226710t^{21} + 176981891328810917456380t^{20} + 275767389418268598707145t^{19} + 299241067534162832251575t^{18} + 240604223659458004001050t^{17} + 148712587883842857192355t^{16} + 72203082271855675247635t^{15} + 27842512947043431969497t^{14} + 8552123524347453878250t^{13} + 2085941671036899584071t^{12} + 401221489106420681912t^{11} + 60387206575645863327t^{10} + 7085594053434609119t^{9} + 654006299354675495t^{8} + 49113319725099363t^{7} + 3191699074040117t^{6} + 190208833075916t^{5} + 10409001903395t^{4} + 485003239495t^{3} + 15089822222t^{2} + 53376033t + 381600 \right)D^{5} + \left( 108835026371724413323500t^{22} + 556730393366691085484450t^{21} + 1325821872691217700062900t^{20} + 1957925359630945089675455t^{19} + 2014905231290891641844840t^{18} + 1537977984114080504427835t^{17} + 903048567741914128909240t^{16} + 416382974710586155442895t^{15} + 152205798347620881707888t^{14} + 44176040586765279196872t^{13} + 10141365605942785073155t^{12} + 1829161004221952953227t^{11} + 257295621954606684384t^{10} + 28003432724002138103t^{9} + 2320163165377504090t^{8} + 138037412066875132t^{7} + 4684087195823056t^{6} - 25663489069857t^{5} - 9908322527122t^{4} - 434730299747t^{3} - 13343225397t^{2} - 41643777t - 190800 \right)D^{4} + t \left( 457107110761242535958700t^{21} + 2238462620913606332398850t^{20} + 5099895935966234626645300t^{19} + 7205569456812386806017555t^{18} + 7099379503565735952943665t^{17} + 5192479889871529047369250t^{16} + 2922209994034680567755605t^{15} + 1290176993212381710846305t^{14} + 450494757617616115299783t^{13} + 124463558336139446198902t^{12} + 27105702918864242723015t^{11} + 4630133894257050098222t^{10} + 618257404331941542647t^{9} + 64449132291680383605t^{8} + 5209027511371229833t^{7} + 314748853689872439t^{6} + 12786666358004507t^{5} + 278367862000500t^{4} + 2328614053159t^{3} + 17035512301t^{2} + 214056250t + 284875 \right)D^{3} + 2 t ^{2} \left( 504994522364801277821040t^{20} + 2388609214329778994707960t^{19} + 5252534446521463992593920t^{18} + 7162683719354192166038950t^{17} + 6814883472369215801738440t^{16} + 4816001749573478447495160t^{15} + 2618420076520153489216830t^{14} + 1115316934175260961346645t^{13} + 374721037326326991809355t^{12} + 99278512328817278495254t^{11} + 20668288639540284283677t^{10} + 3369365631927288284493t^{9} + 429673084023790454129t^{8} + 42855144837609026242t^{7} + 3303806506032732703t^{6} + 186943959063981663t^{5} + 6842072733483788t^{4} + 130568613567313t^{3} + 1415479510406t^{2} + 19499063278t + 94086720 \right)D^{2} + 4 t ^{2} \left( 274264266456745521575220t^{20} + 1263687206799814528251710t^{19} + 2704753653179271683941180t^{18} + 3589364785997610693621315t^{17} + 3324354762349761404786085t^{16} + 2287444174570066749146470t^{15} + 1210399435320452052161945t^{14} + 501005778325701706656860t^{13} + 163148579910913222086074t^{12} + 41766532670485034976433t^{11} + 8379771519220280166671t^{10} + 1314974844682314908215t^{9} + 161527406171884966515t^{8} + 15526270713108606909t^{7} + 1146103100461539496t^{6} + 60654558763509499t^{5} + 1978989081127216t^{4} + 31995775906968t^{3} + 395988006704t^{2} + 5541260936t + 19843200 \right)D + 16 t ^{2} \left( 27986149638443420568900t^{20} + 126633086240553981856950t^{19} + 265965193902119382295500t^{18} + 346226636295136194528375t^{17} + 314562975830601743419325t^{16} + 212322633014748457123525t^{15} + 110143723143185536813400t^{14} + 44628320650990280529025t^{13} + 14193381561013204742110t^{12} + 3539311742531820811570t^{11} + 690157552658861558545t^{10} + 105156816810785418310t^{9} + 12547898552697948023t^{8} + 1171354956855849932t^{7} + 83382280798231885t^{6} + 4161729484623937t^{5} + 122315150516548t^{4} + 1684926944549t^{3} + 22687351470t^{2} + 265667270t + 763200 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × Gr(2, 7)

|λ| = 15.07489 λ ≈ 15.07489; multiplicity 1; Jordan blocks 1 λ ≈ -1.702442 - 9.882025i; multiplicity 1; Jordan blocks 1 λ ≈ -1.702442 + 9.882025i; multiplicity 1; Jordan blocks 1 λ ≈ -2.75149 - 0.8981375i; multiplicity 1; Jordan blocks 1 λ ≈ -2.75149 + 0.8981375i; multiplicity 1; Jordan blocks 1 λ ≈ -2.296539 - 0.3832815i; multiplicity 1; Jordan blocks 1 λ ≈ -2.296539 + 0.3832815i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1.573951; multiplicity 1; Jordan blocks 1 2
Sector dimension
10
Spectral radius
$15.0748936$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$15.07489$11
$-1.702442 - 9.882025\mathrm{i}$11
$-1.702442 + 9.882025\mathrm{i}$11
$-2.75149 - 0.8981375\mathrm{i}$11
$-2.75149 + 0.8981375\mathrm{i}$11
$-2.296539 - 0.3832815\mathrm{i}$11
$-2.296539 + 0.3832815\mathrm{i}$11
$-2$21, 1
$-1.573951$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda^{8} - 32 \lambda^{6} - 1453 \lambda^{5} - 16346 \lambda^{4} - 78640 \lambda^{3} - 190075 \lambda^{2} - 228295 \lambda - 108349\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda^{8} - 32 \lambda^{6} - 1453 \lambda^{5} - 16346 \lambda^{4} - 78640 \lambda^{3} - 190075 \lambda^{2} - 228295 \lambda - 108349\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 6 & 60 & 30 & 33 & 72 & 216 & 352 & 510 \\ 1 & 0 & 5 & -4 & -6 & 4 & 0 & -27 & -42 & 48 \\ 1 & 0 & 0 & 8 & 6 & 2 & 8 & 63 & 108 & 192 \\ 0 & 1 & 1 & 0 & 0 & 3 & 4 & -12 & -24 & -36 \\ 0 & 1 & 0 & 5 & 0 & 3 & 6 & 12 & 20 & 18 \\ 0 & 0 & 1 & 0 & 0 & -2 & 0 & 6 & 12 & 36 \\ 0 & 0 & 1 & 0 & 0 & 0 & -2 & 6 & 12 & 36 \\ 0 & 0 & 0 & -\frac{1}{3} & -\frac{8}{3} & \frac{13}{3} & \frac{10}{3} & -10 & -\frac{40}{3} & -4 \\ 0 & 0 & 0 & \frac{5}{2} & \frac{5}{2} & -1 & 1 & \frac{15}{2} & 10 & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{7}{3} & 0\end{pmatrix}\notag\end{equation}$$