Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-224-51-c

·

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

Defining data
Ambient
Fl(1, 3, 5)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
E^(w1) + 3 copies of E^(w3)
Bundle
O(1,0) + 3 copies of O(0,1)
ZL62 label
44.2220
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,4) \\ \det\mathcal{E} &\cong \mathcal{O}(1,3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 51
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 224
$\chi(X, \mathrm{T}_X)$ = -1

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 224
$\mathrm{c}_1^2\mathrm{c}_2$ = 152
$\mathrm{c}_2^2$ = 97
$\mathrm{c}_1\mathrm{c}_3$ = 54
$\mathrm{c}_4$ = 9
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 3 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 + x1 + 1 0 x0 + 1 x1
$\mathrm{HH}^{ 3 }$ = x1 + 44 0 0 x1 - 1 45
$\mathrm{HH}^{ 4 }$ = 51 0 0 0 0 51
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 46t + 130t^{2} + 46t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(219)
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( 3t + 1 \right) \left( 5283t^{8} - 204t^{7} + 14552t^{6} + 13256t^{5} + 3622t^{4} + 333t^{3} + 29t^{2} + t - 1 \right) \left( 86025982886883t^{13} + 1829227492916100t^{12} + 3903957085745574t^{11} + 3750886935997740t^{10} + 2130289606270587t^{9} + 808859688570808t^{8} + 218550162608965t^{7} + 42360341492150t^{6} + 5711012006011t^{5} + 520871148251t^{4} + 32369309637t^{3} + 1277243747t^{2} + 4820841t + 127456 \right)D^{6} + \left( 28631941858258382007t^{22} + 644642448998098634181t^{21} + 1625489566854187630806t^{20} + 2928046341394314939006t^{19} + 4743014974100734905225t^{18} + 5747846628810744708912t^{17} + 4871227765937754906315t^{16} + 2955637316354853109263t^{15} + 1332751698368273631523t^{14} + 459548782931798107295t^{13} + 123034103630930824751t^{12} + 25693366642038676934t^{11} + 4197117699762148877t^{10} + 545469312371923791t^{9} + 59441719137783630t^{8} + 5945144253419680t^{7} + 574605264623228t^{6} + 49804504616922t^{5} + 3379921125840t^{4} + 163153197945t^{3} + 5123804858t^{2} + 13952699t + 254912 \right)D^{5} + \left( 238599515485486516725t^{22} + 5554581188790613597695t^{21} + 14092590760460901601686t^{20} + 22063071397085450153622t^{19} + 29326826261075039640423t^{18} + 31019855764229902661616t^{17} + 24193223584292795952645t^{16} + 13904737353116422637019t^{15} + 6042061028721857599377t^{14} + 2027759196364553781540t^{13} + 529297598978339702034t^{12} + 106797515729764343249t^{11} + 16424585032633593312t^{10} + 1884464664473550673t^{9} + 153416880457631511t^{8} + 7186277491182368t^{7} - 107053841465325t^{6} - 47789638366902t^{5} - 3641216969473t^{4} - 152135412789t^{3} - 4577787143t^{2} - 9997307t - 127456 \right)D^{4} + t \left( 1002117965039043370245t^{21} + 23931691568481889542735t^{20} + 61223734227014560954170t^{19} + 87367873380768999027090t^{18} + 98274041593988333293107t^{17} + 90322413990184823165928t^{16} + 64378166049646015718421t^{15} + 35030387401619843896449t^{14} + 14743607318763219023873t^{13} + 4858399479263781280483t^{12} + 1252025857697215668067t^{11} + 249680029404836385126t^{10} + 38120733553783398835t^{9} + 4429940170496027589t^{8} + 388891539790882540t^{7} + 24871242296040796t^{6} + 1035736756008316t^{5} + 22943315975454t^{4} + 335918404978t^{3} + 4395952627t^{2} + 122964958t + 100713 \right)D^{3} + 2 t ^{2} \left( 1107101751852657437604t^{20} + 26948035338333006336516t^{19} + 69500349842511927623634t^{18} + 93720435932665923730578t^{17} + 93050073212912108395668t^{16} + 75094574087904681019752t^{15} + 48693883156034574341907t^{14} + 24981493763313879889953t^{13} + 10181412582349491641705t^{12} + 3303268096538318357227t^{11} + 843523336424507619106t^{10} + 166618424456804716479t^{9} + 25136592128402541688t^{8} + 2878993099266982803t^{7} + 247138042896522966t^{6} + 15010911880560897t^{5} + 549451021694970t^{4} + 9091136976955t^{3} + 158768353502t^{2} + 4062257482t + 50779008 \right)D^{2} + 4 t ^{2} \left( 601270779023426022147t^{20} + 14838236989831086796881t^{19} + 38526620256157584975516t^{18} + 50280009587305472599236t^{17} + 45889889883553984949370t^{16} + 33320673182018179312011t^{15} + 19744531325316898310340t^{14} + 9522400881748605039942t^{13} + 3747261923529536866272t^{12} + 1196458876239801357615t^{11} + 303328806064136767458t^{10} + 59575313338864616074t^{9} + 8929518803534535480t^{8} + 1014648480610880946t^{7} + 85807861969069878t^{6} + 4997613971551936t^{5} + 162177539615240t^{4} + 1844232360992t^{3} + 37113442484t^{2} + 1131485904t + 11216128 \right)D + 8 t ^{2} \left( 122708322249678780030t^{20} + 3056154607180802149290t^{19} + 7973532279316817261568t^{18} + 10218771886744894391136t^{17} + 8853410820386713370247t^{16} + 5962701320922781993149t^{15} + 3280070799682486226424t^{14} + 1493316555615294673194t^{13} + 567077267828065857381t^{12} + 178046100393108111900t^{11} + 44844841553528205972t^{10} + 8778088911891741282t^{9} + 1311925605209099316t^{8} + 148579839136448670t^{7} + 12461912703123216t^{6} + 705622603219506t^{5} + 20988265773578t^{4} + 172133149815t^{3} + 4067394838t^{2} + 111671940t + 1019648 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 5)

|λ| = 10.69696 λ ≈ 10.69696; multiplicity 1; Jordan blocks 1 λ ≈ 0.03260414 - 6.699982i; multiplicity 1; Jordan blocks 1 λ ≈ 0.03260414 + 6.699982i; multiplicity 1; Jordan blocks 1 λ ≈ -4.485688; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 λ ≈ -2.770927 - 0.5383872i; multiplicity 1; Jordan blocks 1 λ ≈ -2.770927 + 0.5383872i; multiplicity 1; Jordan blocks 1 λ ≈ 0.1326886 - 0.5387105i; multiplicity 1; Jordan blocks 1 λ ≈ 0.1326886 + 0.5387105i; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$10.6969555$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$10.69696$11
$0.03260414 - 6.699982\mathrm{i}$11
$0.03260414 + 6.699982\mathrm{i}$11
$-4.485688$11
$-3$11
$-2.770927 - 0.5383872\mathrm{i}$11
$-2.770927 + 0.5383872\mathrm{i}$11
$0.1326886 - 0.5387105\mathrm{i}$11
$0.1326886 + 0.5387105\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda^{8} - \lambda^{7} - 29 \lambda^{6} - 333 \lambda^{5} - 3622 \lambda^{4} - 13256 \lambda^{3} - 14552 \lambda^{2} + 204 \lambda - 5283\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda^{8} - \lambda^{7} - 29 \lambda^{6} - 333 \lambda^{5} - 3622 \lambda^{4} - 13256 \lambda^{3} - 14552 \lambda^{2} + 204 \lambda - 5283\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 4 & 9 & 9 & 0 & 4 & 24 & 20 \\ 2 & 0 & 3 & 4 & 0 & 6 & 0 & 0 & 4 \\ 1 & 0 & 0 & 2 & 2 & 0 & 0 & 6 & 8 \\ 0 & 1 & 2 & 0 & 0 & 3 & 0 & 0 & 0 \\ 0 & 2 & 0 & 3 & 0 & 0 & 2 & 4 & 0 \\ 0 & 0 & \frac{5}{3} & 0 & 0 & -2 & 0 & \frac{4}{3} & 3 \\ 0 & 0 & 0 & -6 & -7 & 3 & -3 & -6 & -2 \\ 0 & 0 & 0 & \frac{15}{2} & \frac{9}{2} & 3 & \frac{3}{2} & 3 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 6 & 0\end{pmatrix}\notag\end{equation}$$