Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-55-18-b

·

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

Defining data
Ambient
Fl(1, 4, 5)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)ω1ω2ω3ω4
Bundle
E^(w1) + E^(w1 + w4) + E^(w1 + 2w4)
Bundle
O(1,0) + O(1,1) + O(1,2)
ZL62 label
48.0Ux321
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4,4) \\ \det\mathcal{E} &\cong \mathcal{O}(3,3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 18
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 55
$\chi(X, \mathrm{T}_X)$ = -46

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 55
$\mathrm{c}_1^2\mathrm{c}_2$ = 94
$\mathrm{c}_2^2$ = 166
$\mathrm{c}_1\mathrm{c}_3$ = -14
$\mathrm{c}_4$ = 85
Hodge diamond
1
00
0 2 0
0 0 0 0
0 7 65 7 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = x0 + 46 0 46 x0
$\mathrm{HH}^{ 3 }$ = x0 + 4 0 0 x0 + 2 2
$\mathrm{HH}^{ 4 }$ = 42 0 0 24 0 18
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 13t + 27t^{2} + 13t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(682)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(681)

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) ^{5} \left( 27057t^{4} + 11704t^{3} + 934t^{2} + 9t - 1 \right) \left( 11290584352t^{7} + 9541438432t^{6} + 3852013084t^{5} + 1516213260t^{4} + 453755299t^{3} + 55119612t^{2} + 336260t + 715 \right)D^{6} + \left( 3t + 1 \right) ^{4} \left( 19245828471160032t^{12} + 25916558524935840t^{11} + 16094799515600652t^{10} + 7274723898141660t^{9} + 2785000374750545t^{8} + 764731808803477t^{7} + 123215743593378t^{6} + 9726402831871t^{5} + 273719628294t^{4} + 2445459835t^{3} + 216723948t^{2} + 995910t + 1430 \right)D^{5} + \left( 3t + 1 \right) ^{3} \left( 481145711779000800t^{13} + 706533813780235200t^{12} + 482200874981518164t^{11} + 234887992642728628t^{10} + 95996856387081525t^{9} + 29428658720769502t^{8} + 5733622942827895t^{7} + 621773467460416t^{6} + 31411114787095t^{5} + 520658340694t^{4} + 2972436690t^{3} - 194874605t^{2} - 824035t - 715 \right)D^{4} + t \left( 3t + 1 \right) ^{2} \left( 6062435968415410080t^{13} + 9915693715734772320t^{12} + 7548049090035883620t^{11} + 3985074749165085276t^{10} + 1733518902643695135t^{9} + 588497065864485633t^{8} + 136376231172786643t^{7} + 19349267808987588t^{6} + 1497542226197182t^{5} + 51886492429090t^{4} + 579514398434t^{3} + 4323244591t^{2} + 13302204t + 3510 \right)D^{3} + 2 \left( 3t + 1 \right) t ^{2} \left( 20092644923891073408t^{13} + 37006399364287236048t^{12} + 31678709338264089672t^{11} + 18280505917669817844t^{10} + 8481648838274742594t^{9} + 3156125880070817324t^{8} + 852562377690964452t^{7} + 152179284218243475t^{6} + 16480470669863405t^{5} + 960824742471259t^{4} + 24347030432593t^{3} + 226926873347t^{2} + 2011480305t + 5591080 \right)D^{2} + 4 t ^{2} \left( 32737154229443214432t^{14} + 68157130025787831216t^{13} + 65861396876222739540t^{12} + 41890784648285027916t^{11} + 20850556506648056361t^{10} + 8430928403598216915t^{9} + 2600809462998806514t^{8} + 565043651368163328t^{7} + 80609456192540619t^{6} + 6965995252814425t^{5} + 320702310622070t^{4} + 6321115506908t^{3} + 56610346954t^{2} + 432893075t + 1049620 \right)D + 8 t ^{2} \left( 6681051883559839680t^{14} + 13489071111649095840t^{13} + 12658723051569344424t^{12} + 7887653782358541660t^{11} + 3886789193170978752t^{10} + 1553797597137756582t^{9} + 467569689477158538t^{8} + 97391375667840350t^{7} + 13062527714547258t^{6} + 1034096605846789t^{5} + 41908692691326t^{4} + 695981474017t^{3} + 6384789580t^{2} + 35834955t + 65780 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 4, 5)

|λ| = 40.04433 λ ≈ 40.04433; multiplicity 1; Jordan blocks 1 λ ≈ -14.02007 - 5.324385i; multiplicity 1; Jordan blocks 1 λ ≈ -14.02007 + 5.324385i; multiplicity 1; Jordan blocks 1 λ ≈ -3.004188; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 5; Jordan blocks 1, 1, 3 5
Sector dimension
9
Spectral radius
$40.0443287$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$40.04433$11
$-14.02007 - 5.324385\mathrm{i}$11
$-14.02007 + 5.324385\mathrm{i}$11
$-3.004188$11
$-3$51, 1, 3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right)^{5} \left(\lambda^{4} - 9 \lambda^{3} - 934 \lambda^{2} - 11704 \lambda - 27057\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right)^{3} \left(\lambda^{4} - 9 \lambda^{3} - 934 \lambda^{2} - 11704 \lambda - 27057\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 44 & 48 & 714 & 414 & 687 & 8404 & 6748 & 7625 \\ 1 & -1 & 7 & 78 & 62 & 54 & 1086 & 822 & 884 \\ 1 & 4 & 0 & 108 & 64 & 96 & 1026 & 798 & 996 \\ 0 & 1 & 1 & -1 & 0 & 5 & 78 & 72 & 72 \\ 0 & 1 & 0 & 7 & -1 & 9 & 52 & 48 & 69 \\ 0 & 0 & 1 & 4 & 5 & -2 & 82 & 60 & 54 \\ 0 & 0 & 0 & 1 & -\frac{4}{17} & \frac{35}{17} & -\frac{15}{17} & \frac{62}{17} & \frac{124}{17} \\ 0 & 0 & 0 & 1 & \frac{27}{17} & -\frac{11}{17} & \frac{131}{17} & -\frac{2}{17} & \frac{64}{17} \\ 0 & 0 & 0 & 0 & 0 & 0 & 9 & 7 & 0\end{pmatrix}\notag\end{equation}$$