Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-112-30-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^(2w3) + E^(w1 + 2w3)
Bundle
O_2(2) + O_1(1) box O_2(2)
ZL62 label
2040.8x311
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 30
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 112
$\chi(X, \mathrm{T}_X)$ = -23

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 112
$\mathrm{c}_1^2\mathrm{c}_2$ = 124
$\mathrm{c}_2^2$ = 112
$\mathrm{c}_1\mathrm{c}_3$ = 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 + 23 0 x0 + 23 x2
$\mathrm{HH}^{ 3 }$ = x1 + 15 0 0 x1 15
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 15 0 0 x1 - x2 - 19 4 30
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 25t + 60t^{2} + 25t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(621)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(636)

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( 4t + 1 \right) \left( 6t + 1 \right) \left( 151408t^{6} + 88224t^{5} + 11376t^{4} - 904t^{3} - 36t^{2} + 24t - 1 \right) \left( 16385702585233408t^{13} + 20945057564889088t^{12} + 13881350092546560t^{11} + 8585069063506176t^{10} + 3909973487999088t^{9} + 1040783524588224t^{8} + 153636078446336t^{7} + 12591380389104t^{6} + 673925268672t^{5} + 37028928040t^{4} + 1367258664t^{3} + 21537408t^{2} - 1013623t + 532 \right)D^{6} + \left( 2500773868681219997171712t^{22} + 6384191800585166453997568t^{21} + 7758737517524249617760256t^{20} + 6448422357565674522181632t^{19} + 4245892451635809969254400t^{18} + 2226787037966270815460352t^{17} + 884326633669856543007744t^{16} + 255381893224974340770048t^{15} + 52384907860064593554432t^{14} + 7498738172042263872000t^{13} + 735695013593691149440t^{12} + 48618404955846753024t^{11} + 2119373513784380736t^{10} + 61387125870260624t^{9} + 4437955059128640t^{8} + 874824474236784t^{7} + 85621487295232t^{6} + 4625927483616t^{5} + 226678517832t^{4} + 5780380440t^{3} + 74650092t^{2} - 3028101t + 1064 \right)D^{5} + \left( 20839782239010166643097600t^{22} + 49455060749038880624476160t^{21} + 56680696174375290114342912t^{20} + 46000749883740050304843776t^{19} + 30148155584134098431029248t^{18} + 15514335445412759268572160t^{17} + 5905290848229257358807296t^{16} + 1610505259136362330693952t^{15} + 311189078928226138597120t^{14} + 42502528371767608146688t^{13} + 4135978096564249664192t^{12} + 295473984797372758912t^{11} + 15956929829509109536t^{10} + 516244324224477168t^{9} - 18132569146045456t^{8} - 3953741343982564t^{7} - 265531706095120t^{6} - 9692521587232t^{5} - 323816182512t^{4} - 7190670608t^{3} - 76331672t^{2} + 1925349t - 532 \right)D^{4} + t \left( 87527085403842699901009920t^{21} + 195347681101699527198638080t^{20} + 213493515163661110403399680t^{19} + 170750560938629002718609408t^{18} + 111424879680490702488096768t^{17} + 56017044905347088447407104t^{16} + 20397355028838528673310208t^{15} + 5259164882033035437476224t^{14} + 958793624658859460893184t^{13} + 124864757199743099131904t^{12} + 11968597560680716574208t^{11} + 898711717050760474624t^{10} + 55999985333747609088t^{9} + 2729350094006481968t^{8} + 82381361304379936t^{7} + 241556241030472t^{6} - 106092900053440t^{5} - 3318876440672t^{4} - 54983901800t^{3} - 2533997160t^{2} + 17201908t + 1881 \right)D^{3} + 4 t ^{2} \left( 48348294794503586611986432t^{20} + 102681256244613025910226944t^{19} + 108091550007260065390968832t^{18} + 85734935155893968167956480t^{17} + 55689502938845393259182080t^{16} + 27333232070248911962392064t^{15} + 9550744465986975618952640t^{14} + 2342247344738803332298416t^{13} + 405876544951231052503872t^{12} + 50766977000072239844096t^{11} + 4804975537728023782704t^{10} + 370527620359703761376t^{9} + 24015479189288752728t^{8} + 1180004909372812192t^{7} + 33981720539743188t^{6} + 1950818247497t^{5} - 41155076255104t^{4} - 1169508496948t^{3} - 6544811400t^{2} - 285503558t + 1858976 \right)D^{2} + 8 t ^{2} \left( 26258125621152809970302976t^{20} + 53686973744723130888617984t^{19} + 54939297760560977953374208t^{18} + 43387123201022425587372032t^{17} + 28054215476097425548468224t^{16} + 13475440361411650113162240t^{15} + 4546709167051047055223616t^{14} + 1069759566436419954347920t^{13} + 177907466324065836985664t^{12} + 21578970787339460205216t^{11} + 2028909947982107284176t^{10} + 159394358221384766208t^{9} + 10472161060377758632t^{8} + 505370634412240736t^{7} + 13574258833915052t^{6} - 55459653099965t^{5} - 16923141372480t^{4} - 440305329406t^{3} - 1325874616t^{2} - 17526692t + 171304 \right)D + 16 t ^{2} \left( 5358801147174042851082240t^{20} + 10669800438527360398786560t^{19} + 10705518572273426206261248t^{18} + 8436481749879339338637312t^{17} + 5434484646173429785214976t^{16} + 2566832889529922442359808t^{15} + 843335427140608443385728t^{14} + 192296637680319405318240t^{13} + 31009661121510036189184t^{12} + 3679494328264552267168t^{11} + 344892105982478342560t^{10} + 27418898988727131024t^{9} + 1804551425411528464t^{8} + 85262893945758144t^{7} + 2183934923347080t^{6} - 13031346284574t^{5} - 2628782149184t^{4} - 61132317162t^{3} - 173957828t^{2} + 2648771t + 11704 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ4P4

|λ| = 21.96953 λ ≈ 21.96953; multiplicity 1; Jordan blocks 1 λ ≈ 6.968677 - 7.843974i; multiplicity 1; Jordan blocks 1 λ ≈ 6.968677 + 7.843974i; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 1; Jordan blocks 1 λ ≈ -4.445119; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 1; Jordan blocks 1 λ ≈ -3.73088 - 0.4044034i; multiplicity 1; Jordan blocks 1 λ ≈ -3.73088 + 0.4044034i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$21.9695252$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$21.96953$11
$6.968677 - 7.843974\mathrm{i}$11
$6.968677 + 7.843974\mathrm{i}$11
$-6$11
$-4.445119$11
$-4$11
$-3.73088 - 0.4044034\mathrm{i}$11
$-3.73088 + 0.4044034\mathrm{i}$11
$-2$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 4\right) \left(\lambda + 6\right) \left(\lambda^{6} - 24 \lambda^{5} + 36 \lambda^{4} + 904 \lambda^{3} - 11376 \lambda^{2} - 88224 \lambda - 151408\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 4\right) \left(\lambda + 6\right) \left(\lambda^{6} - 24 \lambda^{5} + 36 \lambda^{4} + 904 \lambda^{3} - 11376 \lambda^{2} - 88224 \lambda - 151408\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 40 & 24 & 24 & 198 & 320 & 320 & 960 \\ 2 & 0 & 6 & 40 & 0 & 60 & 198 & 12 & 80 \\ 1 & 0 & 4 & 2 & 4 & 40 & 36 & 48 & 160 \\ 0 & 1 & 2 & 4 & 0 & 8 & 40 & 0 & 6 \\ 0 & 2 & 0 & 6 & 0 & 2 & 60 & 40 & 96 \\ 0 & 0 & 1 & 0 & 0 & 0 & 2 & 4 & 12 \\ 0 & 0 & 0 & 1 & 0 & \frac{5}{2} & 0 & 0 & 1 \\ 0 & 0 & 0 & 2 & 1 & -\frac{1}{4} & 8 & 4 & \frac{39}{2} \\ 0 & 0 & 0 & 0 & 0 & 0 & 5 & 2 & 0\end{pmatrix}\notag\end{equation}$$