Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-84-24-b

·

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

Defining data
Ambient
Fl(1, 2, 3) × ℙ2P2
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
E^(w1 + w2 + 2w3)
Bundle
O_1(1,1) box O_2(2)
ZL62 label
2022.x420n
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,2;3) \\ \det\mathcal{E} &\cong \mathcal{O}(1,1;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 24
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 84
$\chi(X, \mathrm{T}_X)$ = -31

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 84
$\mathrm{c}_1^2\mathrm{c}_2$ = 108
$\mathrm{c}_2^2$ = 138
$\mathrm{c}_1\mathrm{c}_3$ = 12
$\mathrm{c}_4$ = 54
Hodge diamond
1
00
0 3 0
0 0 0 0
0 3 40 3 0
0 0 0 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = x0 + 31 0 31 x0
$\mathrm{HH}^{ 3 }$ = x0 + 9 0 0 x0 + 6 3
$\mathrm{HH}^{ 4 }$ = 31 0 0 7 0 24
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 19t + 44t^{2} + 19t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(641)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(650)

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( 4t + 1 \right) ^{3} \left( 142t^{3} + 47t^{2} + 7t + 1 \right) \left( 704t^{3} + 160t^{2} + 20t - 1 \right) \left( 11276375814144t^{12} + 13676291088384t^{11} + 396564963328t^{10} - 7776919758080t^{9} - 5815546508096t^{8} - 2116839783456t^{7} - 451688404624t^{6} - 58882827824t^{5} - 4697267668t^{4} - 236954810t^{3} - 8945937t^{2} - 225267t + 1960 \right)D^{6} + \left( 4t + 1 \right) ^{2} \left( 94691245940621180928t^{19} + 173251554516200325120t^{18} + 84383897517363429376t^{17} - 54991535948723847168t^{16} - 99847743966348255232t^{15} - 67853840752766693376t^{14} - 28675448910188275712t^{13} - 8469516513912593408t^{12} - 1833029787272082560t^{11} - 296694226792183232t^{10} - 36146620094014688t^{9} - 3326222080065456t^{8} - 237124811803608t^{7} - 14334893932524t^{6} - 820755134710t^{5} - 40491674859t^{4} - 1343510335t^{3} - 33716781t^{2} - 946148t + 5880 \right)D^{5} + \left( 4t + 1 \right) \left( 3156374864687372697600t^{20} + 6080744288537511198720t^{19} + 3167390635774400921600t^{18} - 2031916430210081456128t^{17} - 3976241208930717351936t^{16} - 2846886963854832132096t^{15} - 1260904504415446777856t^{14} - 388732005526108563968t^{13} - 87544611005004074496t^{12} - 14733166800472166016t^{11} - 1871658474022064576t^{10} - 180130079714879584t^{9} - 13148283641363152t^{8} - 723295780566056t^{7} - 28090609731652t^{6} - 487603747080t^{5} + 21839418663t^{4} + 1662542299t^{3} + 43987104t^{2} + 741573t - 3920 \right)D^{4} + t \left( 53027097726747861319680t^{20} + 109032382047215828336640t^{19} + 63620246041125836554240t^{18} - 34099248619108713889792t^{17} - 78453335483890394202112t^{16} - 60774669535056877895680t^{15} - 28894613396517533024256t^{14} - 9553239113055200821248t^{13} - 2310547081078174205952t^{12} - 419073038880358397184t^{11} - 57744174824070804736t^{10} - 6093781069643566656t^{9} - 497228370051092512t^{8} - 31900682214419136t^{7} - 1638413675168848t^{6} - 65491906647004t^{5} - 1670691808726t^{4} - 5348418707t^{3} + 955989068t^{2} + 13797903t - 26460 \right)D^{3} + 4 t \left( 29291158744298818633728t^{20} + 57530722662599016579072t^{19} + 29008720664159758450688t^{18} - 23778995891891472171008t^{17} - 44004562999341415145472t^{16} - 31728273937749427093504t^{15} - 14214353160512010647552t^{14} - 4424267037438979033600t^{13} - 1002696009275515193984t^{12} - 169452071040002599488t^{11} - 21655789330607696608t^{10} - 2118406634105575728t^{9} - 161306688933692808t^{8} - 9771950535343812t^{7} - 471684124891038t^{6} - 16570271700145t^{5} - 286503641720t^{4} + 4464067857t^{3} + 256379844t^{2} + 1267154t - 1960 \right)D^{2} + 8 t ^{2} \left( 15908129318024358395904t^{19} + 30171858078749448732672t^{18} + 13385411441213158326272t^{17} - 14704135966050706325504t^{16} - 24000116056578987433984t^{15} - 16442290872577009139712t^{14} - 7044347616150360504320t^{13} - 2093035319334522688000t^{12} - 450924169188249095296t^{11} - 72105689257468027840t^{10} - 8692709589840486752t^{9} - 803312810646230800t^{8} - 58269433156623448t^{7} - 3392000541854884t^{6} - 154490505752794t^{5} - 4636917215097t^{4} - 34262223750t^{3} + 2950249747t^{2} + 76426854t + 84224 \right)D + 1920 t ^{3} \left( 27054641697320337408t^{18} + 50079517366476079104t^{17} + 20092404424463319040t^{16} - 26940860471154958336t^{15} - 40822408432104882176t^{14} - 27021930253588739072t^{13} - 11219873987315323904t^{12} - 3224167029067087360t^{11} - 669322994181855616t^{10} - 102731040911447296t^{9} - 11859794747037776t^{8} - 1051623132124536t^{7} - 73772805184692t^{6} - 4177740209710t^{5} - 181266044270t^{4} - 4673091099t^{3} + 10674608t^{2} + 4444482t + 79380 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 3) × ℙ2P2

|λ| = 26.91613 λ ≈ 26.91613; multiplicity 1; Jordan blocks 1 λ ≈ -1.474527 - 5.734048i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1.474527 + 5.734048i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -3.458063 - 3.767907i; multiplicity 1; Jordan blocks 1 λ ≈ -3.458063 + 3.767907i; multiplicity 1; Jordan blocks 1 λ ≈ -4.050946; multiplicity 2; Jordan blocks 1, 1 λ ≈ -4; multiplicity 4; Jordan blocks 1, 3 2 2 2 4
Sector dimension
13
Spectral radius
$26.9161262$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$26.91613$11
$-1.474527 - 5.734048\mathrm{i}$21, 1
$-1.474527 + 5.734048\mathrm{i}$21, 1
$-3.458063 - 3.767907\mathrm{i}$11
$-3.458063 + 3.767907\mathrm{i}$11
$-4.050946$21, 1
$-4$41, 3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{4} \left(\lambda^{3} + 7 \lambda^{2} + 47 \lambda + 142\right)^{2} \left(\lambda^{3} - 20 \lambda^{2} - 160 \lambda - 704\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{3} \left(\lambda^{3} + 7 \lambda^{2} + 47 \lambda + 142\right) \left(\lambda^{3} - 20 \lambda^{2} - 160 \lambda - 704\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 14 & 14 & 20 & 54 & 126 & 54 & 126 & 72 & 352 & 496 & 480 & 2160 \\ 1 & -1 & 2 & 3 & 10 & 16 & 2 & 20 & 12 & 48 & 78 & 63 & 320 \\ 1 & 2 & -1 & 3 & 2 & 20 & 10 & 16 & 12 & 48 & 60 & 72 & 320 \\ 1 & 2 & 2 & 0 & 8 & 14 & 8 & 14 & 8 & 60 & 78 & 72 & 352 \\ 0 & 1 & 2 & 0 & -1 & 3 & 1 & 6 & 3 & 8 & 16 & 18 & 72 \\ 0 & 1 & 0 & 1 & 2 & -1 & 0 & 2 & 1 & 4 & 10 & 4 & 36 \\ 0 & 2 & 1 & 0 & 1 & 6 & -1 & 3 & 3 & 8 & 16 & 18 & 72 \\ 0 & 0 & 1 & 1 & 0 & 2 & 2 & -1 & 1 & 4 & 2 & 8 & 36 \\ 0 & 0 & 0 & 1 & 0 & 2 & 0 & 2 & -2 & 8 & 8 & 6 & 36 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & \frac{7}{4} & \frac{3}{8} & -\frac{3}{8} & -\frac{1}{2} & \frac{15}{4} & \frac{15}{4} & \frac{33}{2} \\ 0 & 0 & 0 & 0 & 1 & 0 & -\frac{1}{2} & \frac{9}{4} & \frac{3}{4} & 1 & -\frac{3}{2} & \frac{3}{2} & 7 \\ 0 & 0 & 0 & 0 & 0 & 3 & 1 & \frac{3}{2} & \frac{3}{2} & 2 & 3 & 0 & 14 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$