Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-42-15-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ4P4
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_6ω1ω2ω3ω4ω5ω6
Bundle
E^(3w3) + E^(2w1 + w3)
Bundle
O_2(3) + O_1(2) box O_2(1)
ZL62 label
2040.Ex317
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 15
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 42
$\chi(X, \mathrm{T}_X)$ = -31

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 42
$\mathrm{c}_1^2\mathrm{c}_2$ = 84
$\mathrm{c}_2^2$ = 153
$\mathrm{c}_1\mathrm{c}_3$ = -6
$\mathrm{c}_4$ = 27
Hodge diamond
1
00
0 2 0
0 5 5 0
0 3 35 3 0
0 5 5 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 + 31 0 x0 + 31 0
$\mathrm{HH}^{ 3 }$ = x1 0 0 x1 0
$\mathrm{HH}^{ 4 }$ = x1 - 2 0 0 x1 - 17 0 15
$\mathrm{HH}^{ 5 }$ = 3 0 0 3 0
$\mathrm{HH}^{ 6 }$ = 0 0 0 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 10t + 20t^{2} + 10t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(726)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(719)

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) ^{2} \left( 8t + 1 \right) ^{4} \left( 10834t^{3} - 431t^{2} + 65t - 1 \right) \left( 4588205172952064t^{10} + 6052134517015040t^{9} + 2830000408266560t^{8} + 692783559587872t^{7} + 104897397253584t^{6} + 10877438390976t^{5} + 817606290788t^{4} + 44167523650t^{3} + 1567119057t^{2} + 27461483t - 111720 \right)D^{6} + \left( 4t + 1 \right) \left( 8t + 1 \right) ^{3} \left( 33404189175008508444672t^{15} + 51646221273273662693376t^{14} + 30413602393198009223168t^{13} + 9750840602503187640832t^{12} + 1990989167756104770432t^{11} + 285077754779135687488t^{10} + 30454371593769554208t^{9} + 2498578600212083024t^{8} + 158185256737803304t^{7} + 7715231640603124t^{6} + 288428339005418t^{5} + 8469362906527t^{4} + 218967286547t^{3} + 4375134477t^{2} + 114649892t - 335160 \right)D^{5} + \left( 8t + 1 \right) ^{2} \left( 8907783780002268918579200t^{17} + 16341819946782320424386560t^{16} + 12102112792955954848931840t^{15} + 5029482625902198752485376t^{14} + 1349594418212557586200064t^{13} + 253723808887625674989056t^{12} + 35328125733093233906304t^{11} + 3781518265301636914624t^{10} + 316685360436854346976t^{9} + 20647876999431566032t^{8} + 1016698668235371112t^{7} + 35256947020483748t^{6} + 724166324810684t^{5} + 2580835737797t^{4} - 318018488315t^{3} - 8812778660t^{2} - 85592717t + 223440 \right)D^{4} + t \left( 8t + 1 \right) \left( 299301535008076235664261120t^{17} + 566145431497079707265925120t^{16} + 435517805923588565533327360t^{15} + 190123907232477649260863488t^{14} + 54215734978104250640173056t^{13} + 10943562049053149243399168t^{12} + 1649401943163631701776640t^{11} + 192404635716777699183872t^{10} + 17697517484624353948352t^{9} + 1282596232429301882720t^{8} + 71669283725317254912t^{7} + 2943210442630316624t^{6} + 80639214248516604t^{5} + 1147723837742678t^{4} - 2195765731029t^{3} - 347387377604t^{2} - 4167409071t + 4700220 \right)D^{3} + 4 t \left( 1322627735654736889030639616t^{18} + 2598341557003560008948908032t^{17} + 2097054924189431582626283520t^{16} + 970369102243253304373477376t^{15} + 296147281290245784720166912t^{14} + 64511387306431764093797888t^{13} + 10563289944714058824104064t^{12} + 1346423786279009133079104t^{11} + 136264949829022848216672t^{10} + 10985155068198576763312t^{9} + 695012099107426386056t^{8} + 33282984847095993380t^{7} + 1123783696033398030t^{6} + 22758389836586477t^{5} + 126381159422900t^{4} - 4835613296647t^{3} - 92969273908t^{2} - 359599142t + 335160 \right)D^{2} + 8 t ^{2} \left( 718323684019382965594226688t^{17} + 1383781826033727568519102464t^{16} + 1083856681667241035490263040t^{15} + 484968336961257624263221248t^{14} + 143030513611517111293286400t^{13} + 30152420480776631050958336t^{12} + 4791114571323467435639936t^{11} + 594164712245340801944512t^{10} + 58529349184300773343968t^{9} + 4575273698699889836496t^{8} + 278015768514579857624t^{7} + 12552362032050551572t^{6} + 384171488912852106t^{5} + 6207987439823813t^{4} - 16374330132502t^{3} - 2422023623397t^{2} - 28833567714t - 21788928 \right)D + 384 t ^{3} \left( 6108194592001555829882880t^{16} + 11609520702274926846935040t^{15} + 8902800089111825513349120t^{14} + 3888950542723751360517120t^{13} + 1119022354435596937401600t^{12} + 230378463589947369559424t^{11} + 35818827014467005056000t^{10} + 4353855733859154860608t^{9} + 420271993002358417632t^{8} + 32079279801950450640t^{7} + 1888504833452882204t^{6} + 81399182715765610t^{5} + 2301917102607324t^{4} + 30128405821886t^{3} - 326994072310t^{2} - 16630253199t - 160625430 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ4P4

|λ| = 60.84282 λ ≈ 60.84282; multiplicity 1; Jordan blocks 1 λ ≈ 2.078591 - 13.18123i; multiplicity 1; Jordan blocks 1 λ ≈ 2.078591 + 13.18123i; multiplicity 1; Jordan blocks 1 λ ≈ -8; multiplicity 4; Jordan blocks 1, 3 λ ≈ -4; multiplicity 2; Jordan blocks 2 4 2
Sector dimension
9
Spectral radius
$60.8428189$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$60.84282$11
$2.078591 - 13.18123\mathrm{i}$11
$2.078591 + 13.18123\mathrm{i}$11
$-8$41, 3
$-4$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 8\right)^{4} \left(\lambda^{3} - 65 \lambda^{2} + 431 \lambda - 10834\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 8\right)^{3} \left(\lambda^{3} - 65 \lambda^{2} + 431 \lambda - 10834\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 52 & 156 & 1980 & 504 & 2916 & 47520 & 16704 & 372960 \\ 1 & 0 & 12 & 156 & 24 & 288 & 3348 & 1080 & 28224 \\ 1 & 3 & 9 & 148 & 44 & 228 & 4086 & 1503 & 33408 \\ 0 & 1 & 1 & 9 & 1 & 30 & 228 & 60 & 1944 \\ 0 & 1 & 0 & 12 & -2 & 0 & 288 & 108 & 1944 \\ 0 & 0 & 1 & 3 & 1 & 0 & 112 & 44 & 1008 \\ 0 & 0 & 0 & 1 & 0 & 3 & 6 & 3 & 104 \\ 0 & 0 & 0 & 1 & 1 & -4 & 18 & 3 & 104 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{1}{2} & 0\end{pmatrix}\notag\end{equation}$$