Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-64-20-a

·

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

Defining data
Ambient
ℙ3P3 × ℙ5P5
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_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w1 + 2w4) + E^(w2 + w4)
Bundle
O_1(1) box O_2(2) + Q_1^v(1) box O_2(1)
ZL62 label
3050.w8x321
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;6) \\ \det\mathcal{E} &\cong \mathcal{O}(3;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 20
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 64
$\chi(X, \mathrm{T}_X)$ = -44

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 64
$\mathrm{c}_1^2\mathrm{c}_2$ = 100
$\mathrm{c}_2^2$ = 157
$\mathrm{c}_1\mathrm{c}_3$ = -10
$\mathrm{c}_4$ = 77
Hodge diamond
1
00
0 2 0
0 0 0 0
0 6 59 6 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 + 44 0 x0 + 44 x2
$\mathrm{HH}^{ 3 }$ = x1 + 4 0 0 x1 4
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 36 0 0 x1 - x2 + 16 0 20
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 15t + 32t^{2} + 15t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(665)
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) ^{3} \left( 132589t^{6} + 153319t^{5} + 64705t^{4} + 11328t^{3} + 649t^{2} + 4t - 1 \right) \left( 730605679463089t^{13} + 1040094918904152t^{12} - 62459151236671t^{11} - 1128543696641145t^{10} - 1098878559881818t^{9} - 554829910287624t^{8} - 173925052928679t^{7} - 36055066395852t^{6} - 5170696024013t^{5} - 543787482621t^{4} - 42477310475t^{3} - 1922198784t^{2} - 9737083t - 29376 \right)D^{6} + \left( 3t + 1 \right) ^{2} \left( 6102827415362884967523t^{20} + 15686374411929104257578t^{19} + 11822574435200678315531t^{18} - 7356371686103624297785t^{17} - 22883210937878942532289t^{16} - 23185774121915242672796t^{15} - 14411232820754911770665t^{14} - 6235092544361943533646t^{13} - 1977125945876238820500t^{12} - 472613646626779980506t^{11} - 86911681243054669750t^{10} - 12520153931234513806t^{9} - 1429523947582925553t^{8} - 127741380702340497t^{7} - 8400866433267680t^{6} - 359813234800893t^{5} - 8831193343012t^{4} - 203726825418t^{3} - 7584167539t^{2} - 28564977t - 58752 \right)D^{5} + \left( 3t + 1 \right) \left( 152570685384072124188075t^{21} + 401157922073462720452275t^{20} + 306807772080403421737619t^{19} - 210820021744530548329113t^{18} - 658981662685782824394119t^{17} - 692765564656810885514021t^{16} - 450747112109009161015576t^{15} - 205808031714030483115943t^{14} - 69486458970547660234318t^{13} - 17860267643090609731099t^{12} - 3568224291414880162175t^{11} - 564091877853869712163t^{10} - 71437548323153918078t^{9} - 7200150572375538458t^{8} - 552008597555009625t^{7} - 29091614534328473t^{6} - 873798129617927t^{5} - 9507622911207t^{4} + 87137773779t^{3} + 6844533958t^{2} + 24319217t + 29376 \right)D^{4} + t \left( 1922390635839308764769745t^{21} + 5280989196546122818194900t^{20} + 4327565451089943047214270t^{19} - 2704029430510625550861443t^{18} - 9486509835521490904680572t^{17} - 10622983221702950077135993t^{16} - 7362284434402138442274806t^{15} - 3598427938293304247266476t^{14} - 1309379936988738180645751t^{13} - 365489367182428416785450t^{12} - 79901534252142134463930t^{11} - 13919383213072338746766t^{10} - 1958251976630580672143t^{9} - 222726020768136631165t^{8} - 19962982260385666653t^{7} - 1321595372845186610t^{6} - 57674356960465450t^{5} - 1376065097939747t^{4} - 13658233541375t^{3} - 99775489967t^{2} - 417032699t - 125307 \right)D^{3} + 2 t ^{2} \left( 2123783940546283968698004t^{20} + 5483971130069105481795660t^{19} + 3807820411974663790651320t^{18} - 3942474433840957930077432t^{17} - 10536774173699655785177270t^{16} - 10920077470272325697670086t^{15} - 7148045209457608530219813t^{14} - 3316096426054104999226748t^{13} - 1146952114788098422919414t^{12} - 304647904031750612829301t^{11} - 63522914830556710719515t^{10} - 10595858505123118380469t^{9} - 1431979524930161428252t^{8} - 156002039079082652717t^{7} - 13193875645709668072t^{6} - 800294874331261625t^{5} - 30604724011184819t^{4} - 604579312724799t^{3} - 5405894679209t^{2} - 51476129921t - 167514240 \right)D^{2} + 4 t ^{2} \left( 1153434381503585258861847t^{20} + 2838959142774460887774240t^{19} + 1697487633939000699094470t^{18} - 2475286795224728700185263t^{17} - 5696879695388717056583787t^{16} - 5591175044702985616243585t^{15} - 3507199928846695770534237t^{14} - 1563331839403556601463261t^{13} - 519849480398068981657251t^{12} - 132844576502980089152158t^{11} - 26701473767600245392918t^{10} - 4306752578902910338705t^{9} - 563732706842684526021t^{8} - 59188624113805531579t^{7} - 4748617525774666407t^{6} - 265680925708716751t^{5} - 8989915598215890t^{4} - 149770735802290t^{3} - 1309849938354t^{2} - 11656458748t - 31138560 \right)D + 24 t ^{2} \left( 78464923911808521011010t^{20} + 186719584539502435453200t^{19} + 98914170240452924286132t^{18} - 182483642833940042588522t^{17} - 385098915121078732175559t^{16} - 364422170263670557946006t^{15} - 221879874435233860926823t^{14} - 96110830092789044741501t^{13} - 31056234978027865201894t^{12} - 7713862931076128240884t^{11} - 1509049418185218039817t^{10} - 237409685431661564965t^{9} - 30325469857098054158t^{8} - 3089998553108701561t^{7} - 237088536637899698t^{6} - 12374638624852464t^{5} - 376362019336625t^{4} - 5431515452331t^{3} - 48768213384t^{2} - 334527666t - 665856 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 4)

|λ| = 34.28542 λ ≈ 34.28542; multiplicity 1; Jordan blocks 1 λ ≈ -11.14076 - 8.066371i; multiplicity 1; Jordan blocks 1 λ ≈ -11.14076 + 8.066371i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 3; Jordan blocks 3 λ ≈ -2.997193; multiplicity 1; Jordan blocks 1 λ ≈ -2.50335 - 0.7439788i; multiplicity 1; Jordan blocks 1 λ ≈ -2.50335 + 0.7439788i; multiplicity 1; Jordan blocks 1 3
Sector dimension
9
Spectral radius
$34.2854176$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$34.28542$11
$-11.14076 - 8.066371\mathrm{i}$11
$-11.14076 + 8.066371\mathrm{i}$11
$-3$33
$-2.997193$11
$-2.50335 - 0.7439788\mathrm{i}$11
$-2.50335 + 0.7439788\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right)^{3} \left(\lambda^{6} - 4 \lambda^{5} - 649 \lambda^{4} - 11328 \lambda^{3} - 64705 \lambda^{2} - 153319 \lambda - 132589\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right)^{3} \left(\lambda^{6} - 4 \lambda^{5} - 649 \lambda^{4} - 11328 \lambda^{3} - 64705 \lambda^{2} - 153319 \lambda - 132589\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 32 & 36 & 246 & 252 & 297 & 2364 & 1796 & 7080 \\ 1 & -1 & 7 & 14 & 30 & 18 & 351 & 264 & 864 \\ 1 & 3 & 0 & 32 & 48 & 38 & 357 & 294 & 1000 \\ 0 & 1 & 2 & 0 & 2 & 3 & 48 & 30 & 144 \\ 0 & 2 & 1 & 7 & -1 & 9 & 38 & 22 & 174 \\ 0 & 0 & 1 & 0 & 3 & -2 & 36 & 26 & 72 \\ 0 & 0 & 0 & 1 & \frac{19}{7} & \frac{6}{7} & 5 & 9 & \frac{152}{7} \\ 0 & 0 & 0 & 1 & -\frac{8}{7} & \frac{10}{7} & -1 & -6 & \frac{48}{7} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 2 & 0\end{pmatrix}\notag\end{equation}$$