Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-51-17-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ5P5
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(w1 + w3) + E^(w2 + 2w3)
Bundle
O_1(1) box O_2(1) + Q_1 box O_2(2)
ZL62 label
2050.x361w1
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;6) \\ \det\mathcal{E} &\cong \mathcal{O}(2;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 17
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 51
$\chi(X, \mathrm{T}_X)$ = -36

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 51
$\mathrm{c}_1^2\mathrm{c}_2$ = 90
$\mathrm{c}_2^2$ = 156
$\mathrm{c}_1\mathrm{c}_3$ = -6
$\mathrm{c}_4$ = 51
Hodge diamond
1
00
0 2 0
0 0 0 0
0 2 41 2 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 + 36 0 x0 + 36 x2
$\mathrm{HH}^{ 3 }$ = x1 + 1 0 0 x1 1
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 16 0 0 x1 - x2 - 1 0 17
$\mathrm{HH}^{ 5 }$ = 1 0 0 1 0
$\mathrm{HH}^{ 6 }$ = 0 0 0 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 12t + 25t^{2} + 12t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(689)
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( 45t - 1 \right) \left( 4t + 1 \right) ^{3} \left( 5t + 1 \right) ^{3} \left( 189t^{2} + 9t + 1 \right) \left( 105496686480t^{7} + 79555546776t^{6} + 21886544641t^{5} + 2836810057t^{4} + 224973834t^{3} + 13485498t^{2} + 75349t + 85 \right)D^{6} + 2 \left( 4t + 1 \right) ^{2} \left( 5t + 1 \right) ^{2} \left( 188422356887604000t^{12} + 201181685099788800t^{11} + 88148690294504490t^{10} + 20913484635755703t^{9} + 3072754949431830t^{8} + 312461363550565t^{7} + 22714451330877t^{6} + 1043811302975t^{5} + 28501741089t^{4} + 415928295t^{3} + 26485805t^{2} + 113406t + 85 \right)D^{5} + \left( 4t + 1 \right) \left( 5t + 1 \right) \left( 62807452295868000000t^{14} + 87194878028366040000t^{13} + 52004654882253842400t^{12} + 17571506871665893260t^{11} + 3767116181485763001t^{10} + 550981155364700175t^{9} + 58161237229374689t^{8} + 4488408612648406t^{7} + 233070477061004t^{6} + 6618742134268t^{5} + 61941552620t^{4} - 711252658t^{3} - 45253277t^{2} - 163963t - 85 \right)D^{4} + 4 t \left( 1318956498213228000000t^{15} + 2290493748914020800000t^{14} + 1760101645448262960000t^{13} + 790937012486432978400t^{12} + 232153112554104091785t^{11} + 47317656744432637861t^{10} + 6988925711793196161t^{9} + 770866689969068231t^{8} + 63739301263401303t^{7} + 3769967121971271t^{6} + 142317318561835t^{5} + 2829242577087t^{4} + 22761517556t^{3} + 82154520t^{2} + 80640t + 150 \right)D^{3} + 4 t ^{2} \left( 2914265786528275200000t^{14} + 4834158400331195400000t^{13} + 3532291197927486570000t^{12} + 1503018390056005242900t^{11} + 416528415233334183285t^{10} + 80142710872682020805t^{9} + 11207890919434220311t^{8} + 1172793738751576411t^{7} + 91068471686556153t^{6} + 4890540060221105t^{5} + 157257843852465t^{4} + 2409916958133t^{3} + 15543856398t^{2} + 130671014t + 249900 \right)D^{2} + 32 t ^{2} \left( 395686949463968400000t^{14} + 633801318018335190000t^{13} + 445383697559953909500t^{12} + 181576533981391732650t^{11} + 48093118680927644775t^{10} + 8844722738145786999t^{9} + 1185380516175586698t^{8} + 118876300929601757t^{7} + 8733021440946765t^{6} + 428719718142129t^{5} + 11846985051005t^{4} + 140536642785t^{3} + 728025268t^{2} + 7332224t + 16405 \right)D + 16 t ^{2} \left( 323009754664464000000t^{14} + 504946427150624400000t^{13} + 345136605713610372000t^{12} + 136423267367970687300t^{11} + 34955984210689533105t^{10} + 6218473297270906080t^{9} + 807565308651311967t^{8} + 78376768568959995t^{7} + 5499182127864477t^{6} + 249836866241139t^{5} + 6042816366713t^{4} + 57225139077t^{3} + 246486902t^{2} + 1954025t + 4420 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ5P5

|λ| = 45 λ ≈ 45; multiplicity 1; Jordan blocks 1 λ ≈ -4.5 - 12.99038i; multiplicity 1; Jordan blocks 1 λ ≈ -4.5 + 12.99038i; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 3; Jordan blocks 1, 2 λ ≈ -4; multiplicity 3; Jordan blocks 1, 2 3 3
Sector dimension
9
Spectral radius
$45$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$45$11
$-4.5 - 12.99038\mathrm{i}$11
$-4.5 + 12.99038\mathrm{i}$11
$-5$31, 2
$-4$31, 2

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{3} \left(\lambda + 5\right)^{3} \left(\lambda - 45\right) \left(\lambda^{2} + 9 \lambda + 189\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 5\right)^{2} \left(\lambda - 45\right) \left(\lambda^{2} + 9 \lambda + 189\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 32 & 72 & 948 & 360 & 1362 & 16640 & 7232 & 89760 \\ 1 & -1 & 10 & 64 & 16 & 128 & 1368 & 624 & 8688 \\ 1 & 2 & 4 & 96 & 48 & 128 & 2010 & 924 & 10848 \\ 0 & 1 & 1 & 3 & -1 & 16 & 76 & 24 & 612 \\ 0 & 1 & 0 & 10 & 0 & 13 & 140 & 56 & 720 \\ 0 & 0 & 1 & 2 & 2 & 0 & 88 & 48 & 540 \\ 0 & 0 & 0 & 1 & 0 & \frac{5}{2} & 2 & 2 & 48 \\ 0 & 0 & 0 & 1 & 1 & -\frac{3}{2} & 13 & 1 & 36 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{3} & \frac{2}{3} & 0\end{pmatrix}\notag\end{equation}$$