Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-75-22-a

·

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

Defining data
Ambient
ℙ2P2 × Gr(2, 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)node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
2 copies of E^(w4) + 2 copies of E^(w1 + w4)
Bundle
2 copies of O_2(1) + 2 copies of O_1(1) box O_2(1)
ZL62 label
2041.33ii
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 22
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 75
$\chi(X, \mathrm{T}_X)$ = -28

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 75
$\mathrm{c}_1^2\mathrm{c}_2$ = 102
$\mathrm{c}_2^2$ = 139
$\mathrm{c}_1\mathrm{c}_3$ = 12
$\mathrm{c}_4$ = 42
Hodge diamond
1
00
0 2 0
0 0 0 0
0 2 32 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 + x1 + 28 0 x0 + 28 x1
$\mathrm{HH}^{ 3 }$ = x1 + 12 0 0 x1 + 10 2
$\mathrm{HH}^{ 4 }$ = 25 0 0 3 0 22
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 17t + 39t^{2} + 17t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(653)
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( 4t + 1 \right) ^{4} \left( 67789t^{6} + 57732t^{5} + 20169t^{4} + 3398t^{3} + 297t^{2} + 15t - 1 \right) \left( 36983307896045t^{13} + 140137233666540t^{12} + 213487194975297t^{11} + 178207262070256t^{10} + 92051349458001t^{9} + 31144471662918t^{8} + 7071328325087t^{7} + 1082739592833t^{6} + 111457171815t^{5} + 7774031939t^{4} + 377140470t^{3} + 11409390t^{2} + 67550t + 150 \right)D^{6} + 3 \left( 4t + 1 \right) ^{3} \left( 70197720851019846140t^{20} + 334576957925147983915t^{19} + 686133852858799814852t^{18} + 822571931458466887812t^{17} + 653733801686407202122t^{16} + 368080060618442921751t^{15} + 152628765989385966867t^{14} + 47706760074581649692t^{13} + 11392982552909572557t^{12} + 2093657062426860982t^{11} + 297097035072730469t^{10} + 32661790489666519t^{9} + 2805702826599525t^{8} + 192228629092630t^{7} + 10819140847365t^{6} + 503277580533t^{5} + 18571176648t^{4} + 560635740t^{3} + 14870770t^{2} + 67050t + 100 \right)D^{5} + \left( 4t + 1 \right) ^{2} \left( 7019772085101984614000t^{21} + 34512702518356504335400t^{20} + 73057411281747336097641t^{19} + 90494963949913515443468t^{18} + 74417226112155261589456t^{17} + 43444017396082779509046t^{16} + 18731052028098351676109t^{15} + 6110393933710232670643t^{14} + 1530392615231493128910t^{13} + 296769124399123543196t^{12} + 44761815862081820134t^{11} + 5265218358446944341t^{10} + 483610585584279517t^{9} + 34530180234608561t^{8} + 1864562530197957t^{7} + 69203447675339t^{6} + 1255202577556t^{5} - 16620372057t^{4} - 1428743990t^{3} - 37483510t^{2} - 147300t - 150 \right)D^{4} + t \left( 4t + 1 \right) \left( 117932171029713341515200t^{21} + 600057285966091577446800t^{20} + 1319161160169927926620900t^{19} + 1702321587525522203687175t^{18} + 1463238945409443291330908t^{17} + 896281128642289552287912t^{16} + 407291114917492144900586t^{15} + 140798760894654712679767t^{14} + 37615856700844413182903t^{13} + 7842381369563361756652t^{12} + 1283635407634552196721t^{11} + 165643703252052551626t^{10} + 16913580550187033175t^{9} + 1369420427486783291t^{8} + 87370494994439539t^{7} + 4262925465082790t^{6} + 147572294528871t^{5} + 3137406477763t^{4} + 32505212478t^{3} + 198125060t^{2} + 840850t + 650 \right)D^{3} + 2 t ^{2} \left( 521147879597971337743360t^{21} + 2750731533783909510751360t^{20} + 6307064663887187371021936t^{19} + 8527661273271646931236672t^{18} + 7714342353371329765638937t^{17} + 4996593799628534627865586t^{16} + 2413547835178833199304839t^{15} + 892189149978217635528892t^{14} + 256625822691319069991110t^{13} + 58052910394769940179143t^{12} + 10399578043373060497129t^{11} + 1482288528930254607700t^{10} + 168728047921575711128t^{9} + 15373579026900236189t^{8} + 1117610931490335398t^{7} + 63617386703670145t^{6} + 2699472119439073t^{5} + 77691925470418t^{4} + 1313226895946t^{3} + 12673985822t^{2} + 147478700t + 498570 \right)D^{2} + 4 t ^{2} \left( 283037210471312019636480t^{21} + 1481499464528228601978240t^{20} + 3352260305174786952090224t^{19} + 4455369269024078222239232t^{18} + 3948156812623292219421601t^{17} + 2496955984167477335424454t^{16} + 1174012942190935094476653t^{15} + 421122455940088958920992t^{14} + 117185184533874526311284t^{13} + 25574158602756310086617t^{12} + 4409630614240887954595t^{11} + 604096518659828630542t^{10} + 66061768345397028624t^{9} + 5780259123052272717t^{8} + 402096279576311266t^{7} + 21629960245295419t^{6} + 842796471467975t^{5} + 21094787804790t^{4} + 284055391780t^{3} + 2406424730t^{2} + 34549550t + 115200 \right)D + 8 t ^{2} \left( 57762696014553473395200t^{21} + 300632699792453697849600t^{20} + 674103948424678252543008t^{19} + 885321938316640226670048t^{18} + 773288860656570613299305t^{17} + 480882004216865401090254t^{16} + 221786327653302506200535t^{15} + 77847643522929022765016t^{14} + 21146051168829206114285t^{13} + 4494533244984928083599t^{12} + 753341541164162903866t^{11} + 100204759352525072996t^{10} + 10635052363476170569t^{9} + 902562316902896620t^{8} + 60674782994049133t^{7} + 3118055070086071t^{6} + 113139635736245t^{5} + 2504986633960t^{4} + 26791463720t^{3} + 190499340t^{2} + 2608250t + 8400 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 5)

|λ| = 29.69953 λ ≈ 29.69953; multiplicity 1; Jordan blocks 1 λ ≈ -2.714511 - 7.126627i; multiplicity 1; Jordan blocks 1 λ ≈ -2.714511 + 7.126627i; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 4; Jordan blocks 2, 2 λ ≈ -3.832189; multiplicity 1; Jordan blocks 1 λ ≈ -2.71916 - 1.687463i; multiplicity 1; Jordan blocks 1 λ ≈ -2.71916 + 1.687463i; multiplicity 1; Jordan blocks 1 4
Sector dimension
10
Spectral radius
$29.6995319$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$29.69953$11
$-2.714511 - 7.126627\mathrm{i}$11
$-2.714511 + 7.126627\mathrm{i}$11
$-4$42, 2
$-3.832189$11
$-2.71916 - 1.687463\mathrm{i}$11
$-2.71916 + 1.687463\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{4} \left(\lambda^{6} - 15 \lambda^{5} - 297 \lambda^{4} - 3398 \lambda^{3} - 20169 \lambda^{2} - 57732 \lambda - 67789\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda^{6} - 15 \lambda^{5} - 297 \lambda^{4} - 3398 \lambda^{3} - 20169 \lambda^{2} - 57732 \lambda - 67789\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 18 & 38 & 312 & 96 & 180 & 276 & 1504 & 1320 & 6880 \\ 1 & -1 & 6 & 32 & 6 & 26 & 40 & 174 & 150 & 896 \\ 1 & 2 & 2 & 42 & 16 & 22 & 34 & 228 & 210 & 1056 \\ 0 & 1 & 1 & 1 & 0 & 4 & 6 & 16 & 10 & 96 \\ 0 & 1 & 0 & 6 & -1 & 1 & 2 & 24 & 20 & 96 \\ 0 & 0 & 1 & 2 & 1 & -3 & 0 & 8 & 10 & 48 \\ 0 & 0 & 1 & 2 & 1 & 2 & 0 & 20 & 20 & 96 \\ 0 & 0 & 0 & \frac{5}{2} & 0 & 3 & \frac{9}{2} & 3 & 5 & 36 \\ 0 & 0 & 0 & 1 & 1 & -\frac{6}{5} & -\frac{9}{5} & \frac{8}{5} & -2 & \frac{8}{5} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{5}{4} & 0\end{pmatrix}\notag\end{equation}$$