Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-100-27-b

·

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

Defining data
Ambient
ℙ6P6 × Gr(2, 4)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8 (marked)node 9: fundamental weight ω_9ω1ω2ω3ω4ω5ω6ω7ω8ω9
Bundle
E^(w8) + 2 copies of E^(w1 + w7) + E^(2w1)
Bundle
O_2(1) + 2 copies of O_1(1) box U_2^v + O_1(2)
ZL62 label
31600.C1w2w2
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(7;4) \\ \det\mathcal{E} &\cong \mathcal{O}(6;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 27
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 100
$\chi(X, \mathrm{T}_X)$ = -20

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 100
$\mathrm{c}_1^2\mathrm{c}_2$ = 112
$\mathrm{c}_2^2$ = 124
$\mathrm{c}_1\mathrm{c}_3$ = 24
$\mathrm{c}_4$ = 24
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 18 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 + x2 + 20 0 x0 + 20 x2
$\mathrm{HH}^{ 3 }$ = x1 + 8 0 0 x1 8
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 11 0 0 x1 - x2 - 16 0 27
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 22t + 54t^{2} + 22t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(600)
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( 2t + 1 \right) ^{2} \left( 3t + 1 \right) ^{2} \left( 4248t^{4} + 2436t^{3} + 290t^{2} + 3t - 1 \right) \left( 13965356552328t^{13} + 52520875787520t^{12} + 82539276873768t^{11} + 70724948950872t^{10} + 36398908806492t^{9} + 11579752601211t^{8} + 2211919415958t^{7} + 217863297840t^{6} + 2297854779t^{5} - 1563249118t^{4} - 87658000t^{3} + 5245380t^{2} + 142027t + 175 \right)D^{6} + \left( 2t + 1 \right) \left( 3t + 1 \right) \left( 7474929163920457344t^{20} + 30725326997392151808t^{19} + 50546243259423747648t^{18} + 38631996121166087040t^{17} + 5729159829491943192t^{16} - 15465365235158470344t^{15} - 15827096588375040624t^{14} - 8122951383402530448t^{13} - 2612615246890998972t^{12} - 544339448611184022t^{11} - 69237062161707456t^{10} - 3910510726442604t^{9} + 203631947474053t^{8} + 42439268775271t^{7} + 761616344527t^{6} - 179124389993t^{5} - 4264735751t^{4} + 317555251t^{3} - 18320603t^{2} - 422231t - 350 \right)D^{5} + \left( 373746458196022867200t^{22} + 1830641096597561688960t^{21} + 3839834653928836282080t^{20} + 4423807047214181980800t^{19} + 2880155743087063105584t^{18} + 772547718546555976800t^{17} - 358173550900916056800t^{16} - 474844938444456939900t^{15} - 248021161423014238104t^{14} - 79873132019424372402t^{13} - 16843937481090085617t^{12} - 2197840836695453742t^{11} - 121741826852299008t^{10} + 11449650269552249t^{9} + 2698973899937709t^{8} + 173416262848166t^{7} - 2254247729056t^{6} - 814300245106t^{5} - 37861200323t^{4} - 926901212t^{3} + 17361664t^{2} + 310731t + 175 \right)D^{4} + t \left( 1569735124423296042240t^{21} + 7632326509872944734080t^{20} + 15977005159044470915520t^{19} + 18710282480633656771392t^{18} + 13146098297431256976240t^{17} + 5181000842031448256640t^{16} + 477902400710331716352t^{15} - 678578264288824321044t^{14} - 441550711358508093264t^{13} - 143868636218285881080t^{12} - 28495109955666020220t^{11} - 3222216457162914198t^{10} - 102026993150767182t^{9} + 23234808200895421t^{8} + 2662585744437796t^{7} - 79076876242732t^{6} - 32225791009432t^{5} - 2090901999522t^{4} - 48916083928t^{3} - 520632354t^{2} - 395530t - 427 \right)D^{3} + 2 t ^{2} \left( 1734183566029546103808t^{20} + 8384260801330898235648t^{19} + 17491921450739406863568t^{18} + 20607439469745934680480t^{17} + 14975197924548136694400t^{16} + 6703271881794061012512t^{15} + 1548552060299164852008t^{14} - 88617131641502660628t^{13} - 200451378963759489924t^{12} - 73337274375606952020t^{11} - 14371586165198844510t^{10} - 1491192739870608066t^{9} - 30253422917429532t^{8} + 9346910077348269t^{7} + 174398543464234t^{6} - 193989274965879t^{5} - 24289514565514t^{4} - 1106904509677t^{3} - 20735930028t^{2} - 178485023t - 370350 \right)D^{2} + 8 t ^{2} \left( 470920537326988812672t^{20} + 2267279619390423163584t^{19} + 4713956974257563875896t^{18} + 5560667685701218477008t^{17} + 4102175524731471097488t^{16} + 1937778781569064907472t^{15} + 550901564295674445756t^{14} + 60423764679071451684t^{13} - 18701066035450284264t^{12} - 9781108649486239158t^{11} - 2029932272871272556t^{10} - 210692175831129951t^{9} - 6160242887014356t^{8} + 252198315380403t^{7} - 204634897657790t^{6} - 50843035316563t^{5} - 4396714036982t^{4} - 158013016697t^{3} - 2619546482t^{2} - 15849334t - 46550 \right)D + 144 t ^{2} \left( 10678470234172081920t^{20} + 51266992771590675840t^{19} + 106299872070058051920t^{18} + 125343856815276067488t^{17} + 93087294550259613048t^{16} + 45081828108032657328t^{15} + 13893824344749378012t^{14} + 2293871207560421544t^{13} - 38392530050837400t^{12} - 116004035329045410t^{11} - 27621603015189486t^{10} - 3113867723475568t^{9} - 168536860495856t^{8} - 19124697509125t^{7} - 6495237234630t^{6} - 1013726974217t^{5} - 70660875898t^{4} - 2130968925t^{3} - 32860784t^{2} - 126427t - 350 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ6P6 × Gr(2, 4)

|λ| = 21.81808 λ ≈ 21.81808; multiplicity 1; Jordan blocks 1 λ ≈ -8.21019 - 3.714297i; multiplicity 1; Jordan blocks 1 λ ≈ -8.21019 + 3.714297i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 2; Jordan blocks 2 λ ≈ -2.397699; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 2 λ ≈ 1; multiplicity 1; Jordan blocks 1 2 2
Sector dimension
9
Spectral radius
$21.8180796$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$21.81808$11
$-8.21019 - 3.714297\mathrm{i}$11
$-8.21019 + 3.714297\mathrm{i}$11
$-3$22
$-2.397699$11
$-2$22
$1$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 3\right)^{2} \left(\lambda - 1\right) \left(\lambda^{4} - 3 \lambda^{3} - 290 \lambda^{2} - 2436 \lambda - 4248\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 3\right)^{2} \left(\lambda - 1\right) \left(\lambda^{4} - 3 \lambda^{3} - 290 \lambda^{2} - 2436 \lambda - 4248\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 20 & 16 & 144 & 156 & 36 & 336 & 896 & 640 \\ 1 & 0 & 2 & 32 & 32 & 8 & 48 & 144 & 112 \\ 1 & 4 & -1 & 20 & 16 & 8 & 72 & 192 & 112 \\ 0 & 1 & 1 & -1 & 4 & 0 & 16 & 32 & 24 \\ 0 & 1 & 0 & 2 & -2 & 1 & 4 & 16 & 6 \\ 0 & 0 & 2 & 8 & 8 & -1 & 4 & 16 & 24 \\ 0 & 0 & 0 & 2 & -2 & 2 & 1 & 12 & 4 \\ 0 & 0 & 0 & 1 & \frac{5}{2} & -\frac{1}{4} & \frac{1}{2} & -2 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 8 & 0\end{pmatrix}\notag\end{equation}$$