Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-320-69-c

·

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

Defining data
Ambient
Fl(1, 2, 5)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
E^(w1) + 2 copies of E^(w2)
Bundle
O(1,0) + 2 copies of O(0,1)
ZL62 label
42.110
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,4) \\ \det\mathcal{E} &\cong \mathcal{O}(1,2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 69
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 320
$\chi(X, \mathrm{T}_X)$ = 4

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 320
$\mathrm{c}_1^2\mathrm{c}_2$ = 176
$\mathrm{c}_2^2$ = 98
$\mathrm{c}_1\mathrm{c}_3$ = 52
$\mathrm{c}_4$ = 10
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 4 0 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 - 4 0 x0 - 4 x1
$\mathrm{HH}^{ 3 }$ = x1 + 56 0 0 x1 - 32 88
$\mathrm{HH}^{ 4 }$ = 69 0 0 0 0 69
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 64t + 190t^{2} + 64t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(102)
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( t + 1 \right) \left( 8t - 1 \right) \left( 1237t^{7} + 1382t^{6} + 495t^{5} + 706t^{4} + 155t^{3} + 65t^{2} + 10t + 1 \right) \left( 200258590786920t^{13} + 13535707258620t^{12} - 446747055360672t^{11} - 723304724584157t^{10} - 477647643143868t^{9} - 174727668165597t^{8} + 39062966516818t^{7} + 19682273774751t^{6} - 4882873063819t^{5} - 367579971773t^{4} + 82085454210t^{3} - 7014225672t^{2} - 32415133t - 1154880 \right)D^{6} + \left( 41616939302974566720t^{22} + 70064553893878850040t^{21} - 64605912591854539056t^{20} - 322849488271158268608t^{19} - 479703129939729827660t^{18} - 423528876273991499828t^{17} - 251102019536650283778t^{16} - 97969031818718430844t^{15} - 28672034377320585316t^{14} - 5157153467346377525t^{13} + 2345604066593699851t^{12} + 751168919592664917t^{11} - 220705496160992975t^{10} - 37811692374538852t^{9} - 7836304167290685t^{8} - 145935939676690t^{7} + 243302808063136t^{6} - 55004560711512t^{5} - 2449358277492t^{4} + 403931743405t^{3} - 28049737929t^{2} - 90316119t - 2309760 \right)D^{5} + \left( 346807827524788056000t^{22} + 479481661712822365800t^{21} - 711548338660921976220t^{20} - 2772779120511391213516t^{19} - 3815492320097150939005t^{18} - 3098897547541999847131t^{17} - 1540871818950221015504t^{16} - 378089423970175958384t^{15} - 40902786937247346255t^{14} + 669455277462775449t^{13} + 11694523937692838175t^{12} + 2065067234640796556t^{11} - 1631047446419036127t^{10} - 207215811650967484t^{9} + 9272372608892004t^{8} - 3126693020786657t^{7} - 118620890131912t^{6} - 13549198517535t^{5} + 3634198198487t^{4} - 485089321613t^{3} + 27967601193t^{2} + 162358679t + 1154880 \right)D^{4} + t \left( 1456592875604109835200t^{21} + 1669337230764499367400t^{20} - 3501140392579699073760t^{19} - 11751659814425954274240t^{18} - 15230828525555533633976t^{17} - 11699827262467356591292t^{16} - 5031374310758901043468t^{15} - 603676015019116447078t^{14} + 173955883361006890838t^{13} + 24456956764254747907t^{12} + 8736223045418910037t^{11} - 3185118047097972239t^{10} - 5223280251847086133t^{9} - 596101583196939000t^{8} + 60540593632991681t^{7} - 8047379379534810t^{6} - 276276505395154t^{5} - 140926103919730t^{4} - 2054847409034t^{3} - 56951764637t^{2} - 667649035t - 133533 \right)D^{3} + 4 t ^{2} \left( 804594159857508289920t^{20} + 776523850455127183920t^{19} - 2132382620800693152732t^{18} - 6489463312738506289792t^{17} - 8035059640461680918011t^{16} - 5954296806883418206812t^{15} - 2268309991148234587905t^{14} - 30741472201335048455t^{13} + 168138530904330337702t^{12} + 620056846067928940t^{11} - 9852303434695034204t^{10} - 4482069979836302057t^{9} - 1951926049782743914t^{8} - 182922703115142390t^{7} + 29751095676025747t^{6} - 4154925275876929t^{5} - 270952485852167t^{4} - 36240688476985t^{3} - 446045620635t^{2} - 17072405711t - 130602880 \right)D^{2} + 4 t ^{2} \left( 873955725362465901120t^{20} + 727579584026258486040t^{19} - 2464089482379036908016t^{18} - 7023252117443260747428t^{17} - 8408310543476773792964t^{16} - 6093365379453407988519t^{15} - 2099578478771433175925t^{14} + 157659361059209873759t^{13} + 205378482986002440643t^{12} - 17745723803890485542t^{11} - 19833299001803968023t^{10} - 5777818808182856596t^{9} - 1418305685141121916t^{8} - 109177442537554822t^{7} + 22996819211121676t^{6} - 4123364017048217t^{5} - 276036486134389t^{4} - 19311945662833t^{3} - 273217702976t^{2} - 10532240116t - 53894400 \right)D + 24 t ^{2} \left( 59452770432820809600t^{20} + 44169208739736901200t^{19} - 174139744424281107840t^{18} - 475943165024253228516t^{17} - 556865057735419625164t^{16} - 398225973044799722031t^{15} - 127069230282159904957t^{14} + 18305829525014243825t^{13} + 14132185554656968505t^{12} - 2231998158767363946t^{11} - 1685933297153715375t^{10} - 401602219803953902t^{9} - 67158888779386890t^{8} - 4230348341796830t^{7} + 1056369000903494t^{6} - 255425083087065t^{5} - 16022224987599t^{4} - 707077530607t^{3} - 13363311788t^{2} - 447540862t - 1539840 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 5)

|λ| = 8 λ ≈ 8; multiplicity 1; Jordan blocks 1 λ ≈ -4.768948 - 4.941334i; multiplicity 1; Jordan blocks 1 λ ≈ -4.768948 + 4.941334i; multiplicity 1; Jordan blocks 1 λ ≈ -0.1330352 - 3.360992i; multiplicity 1; Jordan blocks 1 λ ≈ -0.1330352 + 3.360992i; multiplicity 1; Jordan blocks 1 λ ≈ 0.3382574 - 1.594554i; multiplicity 1; Jordan blocks 1 λ ≈ 0.3382574 + 1.594554i; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 1; Jordan blocks 1 λ ≈ -0.8725477; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$8$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$8$11
$-4.768948 - 4.941334\mathrm{i}$11
$-4.768948 + 4.941334\mathrm{i}$11
$-0.1330352 - 3.360992\mathrm{i}$11
$-0.1330352 + 3.360992\mathrm{i}$11
$0.3382574 - 1.594554\mathrm{i}$11
$0.3382574 + 1.594554\mathrm{i}$11
$-1$11
$-0.8725477$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda - 8\right) \left(\lambda^{7} + 10 \lambda^{6} + 65 \lambda^{5} + 155 \lambda^{4} + 706 \lambda^{3} + 495 \lambda^{2} + 1382 \lambda + 1237\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda - 8\right) \left(\lambda^{7} + 10 \lambda^{6} + 65 \lambda^{5} + 155 \lambda^{4} + 706 \lambda^{3} + 495 \lambda^{2} + 1382 \lambda + 1237\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 2 & 3 & 6 & 6 & 0 & 0 & 5 \\ 1 & -1 & 0 & 2 & 0 & 4 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 & 0 & 0 & 6 & 6 & 0 \\ 0 & 2 & 3 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 3 & 1 & 0 & -1 & 0 & 2 & 4 & 0 \\ 0 & 0 & 2 & 0 & 1 & 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 1 & 5 & 1 & -1 & 0 & 2 \\ 0 & 0 & 0 & 2 & -\frac{3}{2} & \frac{7}{2} & 1 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 5 & 6 & 0\end{pmatrix}\notag\end{equation}$$