Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 4-385-81-a

·

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

Defining data
Ambient
ℙ2P2 × Fl(2, 3, 7)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w6) + E^(w8) + E^(w2 + w3)
Bundle
Q_2,1^v ot U_2,2^v ot wedge^2 U_2,3^v + wedge^3 Q_2,1^v ot U_2,2^v ot wedge^2 U_2,3^v + Q_1 box U_2,3^v
ZL62 label
20605.75w7
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3,5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2,4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 81
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 385
$\chi(X, \mathrm{T}_X)$ = 12

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 385
$\mathrm{c}_1^2\mathrm{c}_2$ = 190
$\mathrm{c}_2^2$ = 100
$\mathrm{c}_1\mathrm{c}_3$ = 62
$\mathrm{c}_4$ = 17
Hodge diamond
1
00
0 4 0
0 0 0 0
0 0 7 0 0
0 0 0 0
0 4 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 - 12 0 x0 - 12 x1
$\mathrm{HH}^{ 3 }$ = x1 + 53 0 0 x1 - 57 110
$\mathrm{HH}^{ 4 }$ = 81 0 0 0 0 81
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 76t + 231t^{2} + 76t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(256)
Coates et al. (2020) Obro(4,38)
Coates–Kasprzyk–Prince (2015) CKP(324)

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( 11t^{4} - 36t^{3} + 8t^{2} + t - 1 \right) \left( 4421t^{5} - 1500t^{4} - 72t^{3} + 75t^{2} + t - 1 \right) \left( 1458247930726620t^{12} + 7821456415573680t^{11} - 10354721748635396t^{10} + 5042046629382735t^{9} - 1594887162293906t^{8} + 381498391512450t^{7} - 55235075056652t^{6} + 3089706238556t^{5} + 123357580584t^{4} - 85349754556t^{3} + 3071748894t^{2} - 184757217t - 1195040 \right)D^{6} + 2 \left( 744618578751245700810t^{21} + 2006742728985821069760t^{20} - 17316339680193537244514t^{19} + 25043894507412384159072t^{18} - 16901271320968165154995t^{17} + 6778540383274937303264t^{16} - 1752103121982104085212t^{15} + 243842357110593221216t^{14} + 24814371357219101242t^{13} - 23348554879861093595t^{12} + 5793595461345660330t^{11} - 663878834336823752t^{10} - 2061815027183490t^{9} + 16249539755737533t^{8} - 3175415099962420t^{7} + 254051495425139t^{6} - 5039322566263t^{5} - 1136818601466t^{4} + 288394059270t^{3} - 7434389035t^{2} + 370709474t + 1792560 \right)D^{5} + \left( 12410309645854095013500t^{21} + 42812147089927846382400t^{20} - 269403742656450098757192t^{19} + 360983576484003237292671t^{18} - 234690846097532150094846t^{17} + 96421346446093962386742t^{16} - 28283187298045461667322t^{15} + 5664283352631285983980t^{14} - 503300512806770989231t^{13} - 84309755337411472691t^{12} + 37136785170867006360t^{11} - 6509237445183242479t^{10} + 658623747778282613t^{9} - 26762955504023670t^{8} + 2642243005772736t^{7} - 777954068809230t^{6} + 31028283691895t^{5} - 362083092949t^{4} - 597143478414t^{3} + 14193492922t^{2} - 798737315t - 2390080 \right)D^{4} + 2 t \left( 26061650256293599528350t^{20} + 105360126540119344515600t^{19} - 527405713306145580063850t^{18} + 659461722696044146093578t^{17} - 413270777340065435033813t^{16} + 169957591085334451715668t^{15} - 51892246154092850670880t^{14} + 11306664234528956849035t^{13} - 1440459145890619205620t^{12} + 33215516816250528047t^{11} + 24966121190679265854t^{10} - 4965030341997871783t^{9} + 248361207205764208t^{8} + 60164313123132279t^{7} - 8864486552278854t^{6} + 553694816056340t^{5} - 37638862795175t^{4} + 78410525476t^{3} + 32026008144t^{2} + 423248636t - 6208 \right)D^{3} + 4 t ^{2} \left( 28791918378381500431320t^{19} + 129460685668610794476660t^{18} - 546120929852648545878571t^{17} + 642215904527748422651505t^{16} - 388775525898358261378025t^{15} + 158349013790431055070388t^{14} - 48708791848431902597064t^{13} + 10792543012584411258951t^{12} - 1479575657143800624923t^{11} + 87473897206625659232t^{10} + 11147602152327806714t^{9} - 3042416169724387223t^{8} + 82552876689484712t^{7} + 50971925744685107t^{6} - 5774082208067404t^{5} + 393499701863609t^{4} - 23014387629605t^{3} + 293712655137t^{2} + 8120174012t + 122460800 \right)D^{2} + 24 t ^{2} \left( 5212330051258719905670t^{19} + 25169840594345689728420t^{18} - 93639830006067816851193t^{17} + 104560786463179690440918t^{16} - 61389985681110314974896t^{15} + 24663980236366635884546t^{14} - 7539655393548456336207t^{13} + 1659952462047514455892t^{12} - 228532057990355618306t^{11} + 15454280386501547742t^{10} + 1326602192150343285t^{9} - 424173596765862362t^{8} + 6916241823900830t^{7} + 6844461992697476t^{6} - 663687482370721t^{5} + 47561968307288t^{4} - 2073511429304t^{3} + 39530015436t^{2} + 551125094t + 4780160 \right)D + 288 t ^{3} \left( 177290137797915643050t^{18} + 895940304825381291300t^{17} - 3059598206939172568021t^{16} + 3284278201595620563180t^{15} - 1881890448183086091329t^{14} + 746079657256468684122t^{13} - 225693991020440239905t^{12} + 49023550168802474464t^{11} - 6627663501883944685t^{10} + 445800989447601785t^{9} + 44177163501770624t^{8} - 13491928925846996t^{7} + 232027271097325t^{6} + 175324934906027t^{5} - 15578081381220t^{4} + 1164472333666t^{3} - 33596697278t^{2} + 849281490t + 15124725 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Fl(2, 3, 7)

|λ| = 7.546885 λ ≈ 7.546885; multiplicity 1; Jordan blocks 1 λ ≈ -6.493476 - 1.695146i; multiplicity 1; Jordan blocks 1 λ ≈ -6.493476 + 1.695146i; multiplicity 1; Jordan blocks 1 λ ≈ -3.799605; multiplicity 2; Jordan blocks 1, 1 λ ≈ 3.220034 - 1.624212i; multiplicity 1; Jordan blocks 1 λ ≈ 3.220034 + 1.624212i; multiplicity 1; Jordan blocks 1 λ ≈ 2.234552 - 1.940705i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 2.234552 + 1.940705i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 0.3305004; multiplicity 2; Jordan blocks 1, 1 2 2 2 2
Sector dimension
13
Spectral radius
$7.54688474$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$7.546885$11
$-6.493476 - 1.695146\mathrm{i}$11
$-6.493476 + 1.695146\mathrm{i}$11
$-3.799605$21, 1
$3.220034 - 1.624212\mathrm{i}$11
$3.220034 + 1.624212\mathrm{i}$11
$2.234552 - 1.940705\mathrm{i}$21, 1
$2.234552 + 1.940705\mathrm{i}$21, 1
$0.3305004$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda^{4} - \lambda^{3} - 8 \lambda^{2} + 36 \lambda - 11\right)^{2} \left(\lambda^{5} - \lambda^{4} - 75 \lambda^{3} + 72 \lambda^{2} + 1500 \lambda - 4421\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda^{4} - \lambda^{3} - 8 \lambda^{2} + 36 \lambda - 11\right) \left(\lambda^{5} - \lambda^{4} - 75 \lambda^{3} + 72 \lambda^{2} + 1500 \lambda - 4421\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 6 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 \\ 1 & 0 & 0 & 0 & 4 & 2 & -2 & 0 & 4 & 0 & -6 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & -2 & 6 & 0 & -4 & 0 & 6 & 3 & 0 \\ 1 & 0 & 0 & 0 & 2 & 2 & -2 & 2 & 6 & 0 & -3 & -3 & 0 \\ 0 & 2 & 3 & 2 & 0 & -1 & -1 & 0 & -1 & 2 & 2 & 2 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 4 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 3 & 3 & 0 & 0 & -2 & 0 & -3 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 2 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 & -2 & 5 & -1 & -7 & 0 & 0 & 0 & -8 \\ 0 & 0 & 0 & 0 & 3 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 7 & 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & \frac{9}{2} & \frac{7}{2} & 0\end{pmatrix}\notag\end{equation}$$