Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-160-39-f

·

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
3 copies of E^(w4) + E^(w1 + w4)
Bundle
3 copies of O_2(1) + O_1(1) box O_2(1)
ZL62 label
2041.333i
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 39
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 160
$\chi(X, \mathrm{T}_X)$ = -9

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 160
$\mathrm{c}_1^2\mathrm{c}_2$ = 136
$\mathrm{c}_2^2$ = 101
$\mathrm{c}_1\mathrm{c}_3$ = 46
$\mathrm{c}_4$ = 13
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 7 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 + 9 0 x0 + 9 x1
$\mathrm{HH}^{ 3 }$ = x1 + 36 0 0 x1 + 15 21
$\mathrm{HH}^{ 4 }$ = 39 0 0 0 0 39
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 34t + 90t^{2} + 34t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(508)
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( 3t + 1 \right) \left( 82981t^{8} + 159433t^{7} + 116154t^{6} + 29829t^{5} + 603t^{4} - 246t^{3} + 97t^{2} + 9t - 1 \right) \left( 851972086002163t^{13} + 1391270750718772t^{12} + 802576266590024t^{11} + 555035990310529t^{10} + 391007065282830t^{9} + 155537305169686t^{8} + 31010863521939t^{7} + 3077867203711t^{6} + 186951146458t^{5} + 16165008899t^{4} + 772466727t^{3} + 10670831t^{2} - 854071t + 544 \right)D^{6} + \left( 4453942227118365737889t^{22} + 15748949690248030502792t^{21} + 24276626860678241692921t^{20} + 23344541856928788326515t^{19} + 17774237451713838745340t^{18} + 12205327008055490121463t^{17} + 7051987477237827362896t^{16} + 3028310163436830628586t^{15} + 899762560971110940699t^{14} + 178041934404191241770t^{13} + 22747605066980576957t^{12} + 1837537127588582473t^{11} + 98438853505388445t^{10} + 5477598273940597t^{9} + 1044929834089875t^{8} + 233348911663311t^{7} + 25035881913603t^{6} + 1407685733923t^{5} + 97301058293t^{4} + 3333054415t^{3} + 37963634t^{2} - 2555685t + 1088 \right)D^{5} + \left( 37116185225986381149075t^{22} + 120798364383873634612840t^{21} + 169596984446896179127937t^{20} + 153834699692871588037614t^{19} + 118327162669304337797130t^{18} + 83067431768298866338538t^{17} + 46825460267118745849169t^{16} + 18960859074760673242347t^{15} + 5267714933206531490640t^{14} + 986359411812523850177t^{13} + 124112273306420891790t^{12} + 10855798858073607189t^{11} + 710005949405989918t^{10} + 22391818619614817t^{9} - 3864499784552653t^{8} - 738361013250834t^{7} - 58722109973177t^{6} - 2514288766598t^{5} - 139930618494t^{4} - 3857364399t^{3} - 42555859t^{2} + 1623397t - 544 \right)D^{4} + t \left( 155887977949142800826115t^{21} + 472891616397231420769520t^{20} + 612679292115903525626455t^{19} + 534042139371837659810893t^{18} + 418896722923135980982544t^{17} + 296294645223536784131833t^{16} + 160883581597758939528696t^{15} + 61243883718879095249014t^{14} + 15915582786275993609881t^{13} + 2820349240441711231580t^{12} + 350278516423313007735t^{11} + 33488759886897017185t^{10} + 2800336871663043801t^{9} + 187935582536302799t^{8} + 7147630641138011t^{7} - 10825148048741t^{6} - 21493196517507t^{5} - 965139352879t^{4} - 21189602887t^{3} - 1149543691t^{2} + 7669942t + 969 \right)D^{3} + 2 t ^{2} \left( 172219099448576808531708t^{20} + 493304458411849867624982t^{19} + 597806996247887412875209t^{18} + 508392497257642271858184t^{17} + 407361169239314234751579t^{16} + 287278597314098438331965t^{15} + 149806267076075327388426t^{14} + 53833814888588810885047t^{13} + 13174857404248136011006t^{12} + 2226085942516995530478t^{11} + 274269304851522538843t^{10} + 27927767696301849116t^{9} + 2556232545962440128t^{8} + 178451735357137713t^{7} + 6757924634630821t^{6} - 21444513796764t^{5} - 16732016689720t^{4} - 761667413797t^{3} - 6118458702t^{2} - 269997906t + 1726848 \right)D^{2} + 4 t ^{2} \left( 93532786769485680495669t^{20} + 256322401871831476071842t^{19} + 294627625386200483073556t^{18} + 247134705795479964847117t^{17} + 201739429613423470536161t^{16} + 141120748104048943225436t^{15} + 70916871014957458908948t^{14} + 24258538217592928245111t^{13} + 5646973648774629982304t^{12} + 919380906184388835750t^{11} + 113255882225528811270t^{10} + 12104604449339022346t^{9} + 1154375372044320329t^{8} + 80149688165443570t^{7} + 2923573378516588t^{6} - 15795724185117t^{5} - 6434457909012t^{4} - 285150991322t^{3} - 1200373692t^{2} - 17769872t + 178432 \right)D + 48 t ^{2} \left( 3181387305084546955635t^{20} + 8452048965839677258130t^{19} + 9350875896366572363866t^{18} + 7788236417112682741123t^{17} + 6446867231956072117523t^{16} + 4471027276754443014998t^{15} + 2181926241633176144748t^{14} + 718380709625573923677t^{13} + 160897909785467351480t^{12} + 25495195382620026258t^{11} + 3152079759657254514t^{10} + 348501230538466258t^{9} + 33782001772180151t^{8} + 2306722480449058t^{7} + 81943598450800t^{6} - 443655549423t^{5} - 159381062982t^{4} - 6577032206t^{3} - 19897224t^{2} + 408850t + 2176 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 5)

|λ| = 15.20256 λ ≈ 15.20256; multiplicity 1; Jordan blocks 1 λ ≈ 4.489248 - 5.410326i; multiplicity 1; Jordan blocks 1 λ ≈ 4.489248 + 5.410326i; multiplicity 1; Jordan blocks 1 λ ≈ -5.669766; multiplicity 1; Jordan blocks 1 λ ≈ -4.465235; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 λ ≈ -2.910546; multiplicity 1; Jordan blocks 1 λ ≈ -1.067754 - 0.5988812i; multiplicity 1; Jordan blocks 1 λ ≈ -1.067754 + 0.5988812i; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$15.2025601$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$15.20256$11
$4.489248 - 5.410326\mathrm{i}$11
$4.489248 + 5.410326\mathrm{i}$11
$-5.669766$11
$-4.465235$11
$-3$11
$-2.910546$11
$-1.067754 - 0.5988812\mathrm{i}$11
$-1.067754 + 0.5988812\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda^{8} - 9 \lambda^{7} - 97 \lambda^{6} + 246 \lambda^{5} - 603 \lambda^{4} - 29829 \lambda^{3} - 116154 \lambda^{2} - 159433 \lambda - 82981\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda^{8} - 9 \lambda^{7} - 97 \lambda^{6} + 246 \lambda^{5} - 603 \lambda^{4} - 29829 \lambda^{3} - 116154 \lambda^{2} - 159433 \lambda - 82981\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 20 & 9 & 9 & 30 & 40 & 80 & 150 \\ 2 & 0 & 3 & 20 & 0 & 12 & 30 & 0 & 8 \\ 1 & 0 & 2 & 2 & 2 & 8 & 6 & 15 & 32 \\ 0 & 1 & 2 & 2 & 0 & 2 & 8 & 0 & 0 \\ 0 & 2 & 0 & 3 & 0 & 0 & 12 & 20 & 30 \\ 0 & 0 & \frac{5}{2} & 0 & 0 & 0 & 2 & 5 & 9 \\ 0 & 0 & 0 & \frac{5}{2} & 0 & 3 & 0 & 0 & 2 \\ 0 & 0 & 0 & 2 & 1 & -\frac{2}{5} & 2 & 2 & \frac{36}{5} \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & \frac{5}{2} & 0\end{pmatrix}\notag\end{equation}$$