Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-86-24-a

·

This is a Fano fourfold of K3 type, appearing as case C-7 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ4P4 × ℙ5P5
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9ω1ω2ω3ω4ω5ω6ω7ω8ω9
Bundle
E^(w1 + w5) + E^(w2 + w5)
Bundle
O_1(1) box O_2(1) + Q_1^v(1) box O_2(1)
ZL62 label
4050.wAw3
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;6) \\ \det\mathcal{E} &\cong \mathcal{O}(4;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 24
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 86
$\chi(X, \mathrm{T}_X)$ = -24

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 86
$\mathrm{c}_1^2\mathrm{c}_2$ = 104
$\mathrm{c}_2^2$ = 136
$\mathrm{c}_1\mathrm{c}_3$ = 16
$\mathrm{c}_4$ = 34
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 26 1 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 + 24 0 x0 + 24 x2
$\mathrm{HH}^{ 3 }$ = x1 + 6 0 0 x1 6
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 15 0 0 x1 - x2 - 9 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 19t + 46t^{2} + 19t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(614)
Coates et al. (2020) Str(3)
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( 2t + 1 \right) \left( t + 1 \right) ^{4} \left( 6824t^{4} + 4084t^{3} + 396t^{2} - t - 1 \right) \left( 4687509344t^{9} + 12374456704t^{8} + 10788543552t^{7} + 3245055360t^{6} + 82507466t^{5} - 51316964t^{4} + 32833275t^{3} + 5085837t^{2} + 39032t + 172 \right)D^{6} + \left( t + 1 \right) ^{3} \left( 1343477678065152t^{15} + 5485958401307648t^{14} + 9063261757147264t^{13} + 7752997138928704t^{12} + 3617698235601728t^{11} + 859462572681024t^{10} + 77746395288264t^{9} + 10398623188652t^{8} + 9781308525218t^{7} + 2736791121262t^{6} + 280465631157t^{5} + 9421822240t^{4} + 208457015t^{3} + 20274348t^{2} + 114344t + 344 \right)D^{5} + \left( t + 1 \right) ^{2} \left( 11195647317209600t^{16} + 51045863957655040t^{15} + 96060018598032768t^{14} + 96086449100099648t^{13} + 54489324986651392t^{12} + 16915443843685376t^{11} + 2359177592320632t^{10} + 132635837994988t^{9} + 131496334368114t^{8} + 63427594276160t^{7} + 11124628351299t^{6} + 815996428042t^{5} + 20790091402t^{4} + 140573110t^{3} - 18194583t^{2} - 101272t - 172 \right)D^{4} + t \left( t + 1 \right) \left( 47021718732280320t^{16} + 242056206231941120t^{15} + 524205532944739200t^{14} + 618781327321632192t^{13} + 429450634181540544t^{12} + 174018793488557696t^{11} + 36931828897335000t^{10} + 3003294318919988t^{9} + 714696493029286t^{8} + 624883030465818t^{7} + 183847694542285t^{6} + 23119817010184t^{5} + 1249282142724t^{4} + 20014241330t^{3} + 236300633t^{2} + 1323756t + 332 \right)D^{3} + 2 t ^{2} \left( 51947803551852544t^{16} + 303000040678368256t^{15} + 756498308692817920t^{14} + 1052868907014187904t^{13} + 889094882510775040t^{12} + 461400538838770976t^{11} + 139401919490824496t^{10} + 20523735682754408t^{9} + 1588206351581072t^{8} + 1262353549303134t^{7} + 648246273097497t^{6} + 136424089965445t^{5} + 13305778501404t^{4} + 555848903010t^{3} + 7682587317t^{2} + 110597095t + 514538 \right)D^{2} + 8 t ^{2} \left( 14106515619684096t^{16} + 79024116815548672t^{15} + 188265636078326592t^{14} + 247882521936954400t^{13} + 195590292157725376t^{12} + 92877477496653072t^{11} + 24525933424889228t^{10} + 2743721743110094t^{9} + 235914501166855t^{8} + 307156401809365t^{7} + 132733986409100t^{6} + 23808182477214t^{5} + 1985115350854t^{4} + 67556603635t^{3} + 935122781t^{2} + 12641134t + 48504 \right)D + 32 t ^{2} \left( 1439440369355520t^{16} + 7839217148125440t^{15} + 18059321022613056t^{14} + 22825608001545504t^{13} + 17101884481074624t^{12} + 7563186960856688t^{11} + 1774430030488268t^{10} + 146363497534782t^{9} + 18891177441755t^{8} + 28799045659575t^{7} + 10826347095817t^{6} + 1712337878069t^{5} + 125446917687t^{4} + 3559364374t^{3} + 53502879t^{2} + 588094t + 1720 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ4P4 × ℙ5P5

|λ| = 23.61313 λ ≈ 23.61313; multiplicity 1; Jordan blocks 1 λ ≈ -11.25988 - 3.356498i; multiplicity 1; Jordan blocks 1 λ ≈ -11.25988 + 3.356498i; multiplicity 1; Jordan blocks 1 λ ≈ -2.093369; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 4; Jordan blocks 2, 2 4
Sector dimension
9
Spectral radius
$23.6131333$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$23.61313$11
$-11.25988 - 3.356498\mathrm{i}$11
$-11.25988 + 3.356498\mathrm{i}$11
$-2.093369$11
$-2$11
$-1$42, 2

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{4} \left(\lambda + 2\right) \left(\lambda^{4} + \lambda^{3} - 396 \lambda^{2} - 4084 \lambda - 6824\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda + 2\right) \left(\lambda^{4} + \lambda^{3} - 396 \lambda^{2} - 4084 \lambda - 6824\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 20 & 20 & 156 & 102 & 168 & 1180 & 1060 & 3060 \\ 1 & -1 & 4 & 24 & 28 & 12 & 240 & 180 & 456 \\ 1 & 2 & -1 & 42 & 26 & 46 & 192 & 174 & 648 \\ 0 & 1 & 1 & -2 & -2 & 0 & 12 & 14 & 36 \\ 0 & 1 & 0 & 4 & 0 & 7 & 24 & 28 & 90 \\ 0 & 0 & 1 & 2 & 4 & -1 & 40 & 30 & 54 \\ 0 & 0 & 0 & 1 & -\frac{9}{25} & \frac{11}{5} & -\frac{53}{25} & \frac{34}{25} & \frac{72}{5} \\ 0 & 0 & 0 & 1 & \frac{47}{25} & -\frac{3}{5} & \frac{124}{25} & \frac{3}{25} & \frac{24}{5} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{13}{6} & \frac{11}{6} & 0\end{pmatrix}\notag\end{equation}$$