Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 5-209-48-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ4P4
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w4 + w5) + E^(w3 + w5) + E^(w2 + w5) + E^(w1 + w5)
Bundle
O_4(1) box O_5(1) + O_3(1) box O_5(1) + O_2(1) box O_5(1) + O_1(1) box O_5(1)
ZL62 label
111140.wIwEw9w3
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;2;2;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;1;1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 48
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 209
$\chi(X, \mathrm{T}_X)$ = 0

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 209
$\mathrm{c}_1^2\mathrm{c}_2$ = 146
$\mathrm{c}_2^2$ = 102
$\mathrm{c}_1\mathrm{c}_3$ = 62
$\mathrm{c}_4$ = 23
Hodge diamond
1
00
0 5 0
0 0 0 0
0 0 11 0 0
0 0 0 0
0 5 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 0 x0 x1
$\mathrm{HH}^{ 3 }$ = x1 + 26 0 0 x1 - 6 32
$\mathrm{HH}^{ 4 }$ = 48 0 0 0 0 48
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 43t + 121t^{2} + 43t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(408)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(476)

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( 59t^{3} + 13t^{2} - 4t - 1 \right) \left( 6277t^{5} + 4056t^{4} + 915t^{3} + 73t^{2} - 3t - 1 \right) \left( 12903466873753480t^{13} + 30924938585234392t^{12} + 32376835471293782t^{11} + 19601122928221218t^{10} + 7642159376873374t^{9} + 2028648457919822t^{8} + 383494233778965t^{7} + 54956227386157t^{6} + 6402439963163t^{5} + 591562064975t^{4} + 35222624412t^{3} + 850036764t^{2} - 3447093t + 147345 \right)D^{6} + \left( 200705762561912371832880t^{22} + 725949525354137761925544t^{21} + 1196186334233612820732564t^{20} + 1188685606847535382245786t^{19} + 793447355512219835088770t^{18} + 374037807667207380240444t^{17} + 126858173864330077239410t^{16} + 30602606518490190515005t^{15} + 4879198629120227078623t^{14} + 336357588399483865400t^{13} - 68759102011500460894t^{12} - 30517997644740880405t^{11} - 6874504091954447818t^{10} - 1151752551344491779t^{9} - 150104953830129865t^{8} - 14950639049024904t^{7} - 1137800811375404t^{6} - 71311500969071t^{5} - 4045015475577t^{4} - 177219917010t^{3} - 3484215093t^{2} + 12993489t - 294690 \right)D^{5} + \left( 1672548021349269765274000t^{22} + 5854009632846228660733880t^{21} + 9339363636635787660261396t^{20} + 9007790041178284926396482t^{19} + 5867645522596191008455348t^{18} + 2729166063705068936845458t^{17} + 934092972739806612096370t^{16} + 238968067874410316233053t^{15} + 45935447441060842164002t^{14} + 6561976432379769089295t^{13} + 645879956998380769736t^{12} + 23402387786538351951t^{11} - 6639264040593932199t^{10} - 1766186076972829995t^{9} - 230706731370912252t^{8} - 17351015193740237t^{7} - 566328507542084t^{6} + 21651289872643t^{5} + 3507996055456t^{4} + 186997526235t^{3} + 4018328001t^{2} - 109266t + 147345 \right)D^{4} + t \left( 7024701689666933014150800t^{21} + 23941460299107926502551640t^{20} + 37176534636741559467717100t^{19} + 34927049312792460040067382t^{18} + 22215666485971764845745094t^{17} + 10142043884790656480317548t^{16} + 3441884814857425469480402t^{15} + 890176952705739489436959t^{14} + 179335040021132774159533t^{13} + 28682141071520872936444t^{12} + 3633418927143708018768t^{11} + 334496451797403847619t^{10} + 13444498116644564212t^{9} - 1860556645550332855t^{8} - 410344467797621373t^{7} - 37212762856029940t^{6} - 1804383744699342t^{5} - 44587042776037t^{4} - 662986117437t^{3} - 20903659920t^{2} - 405698247t - 154395 \right)D^{3} + 2 t ^{2} \left( 7760622819060611710871360t^{20} + 25904113386656344934370424t^{19} + 39355889194695447611379716t^{18} + 36170512908691532803218574t^{17} + 22525844736546228356265478t^{16} + 10093657026894210180875730t^{15} + 3379869524907955654409468t^{14} + 871085887720811818459513t^{13} + 177784973008378483968899t^{12} + 29487741408892345804037t^{11} + 3995225614336112202899t^{10} + 420126260494099212420t^{9} + 28715303756837541651t^{8} + 381376897369919762t^{7} - 152446160552631763t^{6} - 16410619493397211t^{5} - 771690277631956t^{4} - 18408827081760t^{3} - 393086547906t^{2} - 13628764200t - 145891614 \right)D^{2} + 4 t ^{2} \left( 4214821013800159808490480t^{20} + 13851505598564342593836144t^{19} + 20696143174619729943439904t^{18} + 18694948620580516901440644t^{17} + 11443801678113967601244744t^{16} + 5045898287684186851133142t^{15} + 1667483487851245710295934t^{14} + 426625631643731152923326t^{13} + 87263934668060067278608t^{12} + 14675607965348349530451t^{11} + 2038310696421674889229t^{10} + 223362970381131757112t^{9} + 17072234951120324244t^{8} + 628959555779068735t^{7} - 23543966362000292t^{6} - 4016064231572810t^{5} - 190894197878974t^{4} - 4619599048461t^{3} - 126183263859t^{2} - 4044887589t - 28290240 \right)D + 48 t ^{2} \left( 143361258972794551309200t^{20} + 466150622099852145488160t^{19} + 688429646738334803363288t^{18} + 614240902056853661765760t^{17} + 371290377411676342604348t^{16} + 161728775733241817324046t^{15} + 52883844036385670761898t^{14} + 13436755977945611693228t^{13} + 2745596451409287062638t^{12} + 464356651771777013923t^{11} + 65167066906944982977t^{10} + 7250410893779083892t^{9} + 575486872968054918t^{8} + 25702904068525393t^{7} - 32009675849994t^{6} - 71206877475992t^{5} - 3607174898978t^{4} - 95430517219t^{3} - 3064385713t^{2} - 81982335t - 392920 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ4P4

|λ| = 12.02694 λ ≈ 12.02694; multiplicity 1; Jordan blocks 1 λ ≈ -3.862837 - 4.735418i; multiplicity 1; Jordan blocks 1 λ ≈ -3.862837 + 4.735418i; multiplicity 1; Jordan blocks 1 λ ≈ -3.864516 - 0.9419768i; multiplicity 4; Jordan blocks 1, 1, 1, 1 λ ≈ -3.864516 + 0.9419768i; multiplicity 4; Jordan blocks 1, 1, 1, 1 λ ≈ -3.650631 - 0.8050115i; multiplicity 1; Jordan blocks 1 λ ≈ -3.650631 + 0.8050115i; multiplicity 1; Jordan blocks 1 λ ≈ 3.729032; multiplicity 4; Jordan blocks 1, 1, 1, 1 λ ≈ -2; multiplicity 6; Jordan blocks 1, 1, 1, 1, 1, 1 4 4 4 6
Sector dimension
23
Spectral radius
$12.0269371$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$12.02694$11
$-3.862837 - 4.735418\mathrm{i}$11
$-3.862837 + 4.735418\mathrm{i}$11
$-3.864516 - 0.9419768\mathrm{i}$41, 1, 1, 1
$-3.864516 + 0.9419768\mathrm{i}$41, 1, 1, 1
$-3.650631 - 0.8050115\mathrm{i}$11
$-3.650631 + 0.8050115\mathrm{i}$11
$3.729032$41, 1, 1, 1
$-2$61, 1, 1, 1, 1, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{6} \left(\lambda^{3} + 4 \lambda^{2} - 13 \lambda - 59\right)^{4} \left(\lambda^{5} + 3 \lambda^{4} - 73 \lambda^{3} - 915 \lambda^{2} - 4056 \lambda - 6277\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda^{3} + 4 \lambda^{2} - 13 \lambda - 59\right) \left(\lambda^{5} + 3 \lambda^{4} - 73 \lambda^{3} - 915 \lambda^{2} - 4056 \lambda - 6277\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 2 & 2 & 2 & 8 & 3 & 3 & 3 & 9 & 3 & 3 & 9 & 3 & 9 & 9 & 18 & 4 & 4 & 8 & 4 & 4 & 5 \\ 1 & -1 & 0 & 0 & 0 & 1 & 2 & 2 & 2 & 6 & 0 & 0 & 2 & 0 & 2 & 2 & 6 & 3 & 3 & 6 & 3 & 0 & 4 \\ 1 & 0 & -1 & 0 & 0 & 1 & 2 & 0 & 0 & 2 & 2 & 2 & 6 & 0 & 2 & 2 & 6 & 3 & 3 & 6 & 0 & 3 & 4 \\ 1 & 0 & 0 & -1 & 0 & 1 & 0 & 2 & 0 & 2 & 2 & 0 & 2 & 2 & 6 & 2 & 6 & 3 & 0 & 3 & 3 & 3 & 4 \\ 1 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 2 & 2 & 0 & 2 & 2 & 2 & 2 & 6 & 6 & 0 & 3 & 3 & 3 & 3 & 4 \\ 1 & 1 & 1 & 1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 & -2 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 2 & 2 & 4 & 0 & 0 & 3 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & -2 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 0 & 2 & 2 & 0 & 3 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 2 & 2 & 2 & 0 & 3 \\ 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 1 & 0 & 1 & 0 & 1 & 2 & 0 & 2 & 0 & 2 & 3 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 1 & 0 & 0 & 1 & 1 & 0 & 2 & 2 & 0 & 2 & 3 \\ 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 1 & 1 & 1 & 0 & 0 & 0 & 2 & 2 & 3 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & -1 & -2 & 1 & -1 & -2 & -1 & -2 & -5 & -7 & -3 & -2 & -1 & -2 & -2 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 1 & -2 & -1 & 1 & -2 & -1 & -5 & -2 & -7 & -2 & -3 & -1 & -2 & -2 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 5 & 1 & 1 & 5 & 1 & 5 & 5 & 11 & 3 & 3 & 1 & 3 & 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 3 & 0 & 0 & 0 & 2 & 3 & 3 & 4 & 1 & 1 & 1 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 3 & 2 & 3 & 3 & 4 & 1 & 1 & 1 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 3 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$