Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-126-32-b

·

Blow-up of the genus-eight Mukai fourfold along a del Pezzo surface of degree $6$.

  • $\pi_1\colon X\to Y$ is the blow-up of a Fano fourfold $Y$ of index $2$, degree $14$ and genus $8$ (1-224-51-a), a linear section of $\mathrm{Gr}(2,6)$, along a sextic del Pezzo surface, the intersection of two divisors of bidegree $(1,1)$ in $\mathbb P^2\times\mathbb P^2$ (2-486-100-a).

  • $\pi_2\colon X\to\mathbb P^3$ is a conic bundle outside two points, with discriminant a quartic surface with $8$ nodes, while the fibers over the two points are planes.

Varieties in this family can be blown down to

Defining data
Ambient
Fl(1, 4, 6)
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_5ω1ω2ω3ω4ω5
Bundle
2 copies of E^(w1) + E^(w3) + 2 copies of E^(w4)
Bundle
2 copies of O(1,0) + wedge^2 U_2^v ot U_3^v + 2 copies of O(0,1)
ZL62 label
58.33200
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4,5) \\ \det\mathcal{E} &\cong \mathcal{O}(3,4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 32
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 126
$\chi(X, \mathrm{T}_X)$ = -13

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 126
$\mathrm{c}_1^2\mathrm{c}_2$ = 120
$\mathrm{c}_2^2$ = 116
$\mathrm{c}_1\mathrm{c}_3$ = 36
$\mathrm{c}_4$ = 18
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 12 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 + 13 0 x0 + 13 x1
$\mathrm{HH}^{ 3 }$ = x1 + 28 0 0 x1 + 15 13
$\mathrm{HH}^{ 4 }$ = 32 0 0 0 0 32
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 27t + 70t^{2} + 27t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) No match
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( 2t + 1 \right) ^{3} \left( 8t^{2} + 4t + 1 \right) \left( 1712t^{4} + 1184t^{3} + 102t^{2} + 7t - 1 \right) \left( 2667158175744t^{15} + 32605770596352t^{14} + 86768527162368t^{13} + 106937683670016t^{12} + 68828436478080t^{11} + 17003790323584t^{10} - 8079938364224t^{9} - 9138327067200t^{8} - 4038794454296t^{7} - 1058032622072t^{6} - 176349912172t^{5} - 19201905320t^{4} - 1417627502t^{3} - 65376192t^{2} - 447301t - 3339 \right)D^{6} + \left( 2t + 1 \right) ^{2} \left( 4602704195248717824t^{23} + 65834387576362893312t^{22} + 257431963183654109184t^{21} + 515870983365013536768t^{20} + 632016265995240112128t^{19} + 497898616300240404480t^{18} + 235796888223531954176t^{17} + 31931960078776606720t^{16} - 46535163962073685504t^{15} - 44707804033382118400t^{14} - 22946758825372436096t^{13} - 8152989856611769600t^{12} - 2142912784983188960t^{11} - 426980717101665344t^{10} - 65209122261171656t^{9} - 7700075968147568t^{8} - 711578197479876t^{7} - 52121325494120t^{6} - 3040571334726t^{5} - 145997617372t^{4} - 6677849909t^{3} - 257867008t^{2} - 1288479t - 6678 \right)D^{5} + \left( 2t + 1 \right) \left( 76711736587478630400t^{24} + 1138883956593131520000t^{23} + 4639868505518101757952t^{22} + 9710034054751078318080t^{21} + 12452849148268277022720t^{20} + 10298482203600799531008t^{19} + 5143630266630145042432t^{18} + 756812599308976365568t^{17} - 1106270955745806511616t^{16} - 1137489339294081773568t^{15} - 624322960008293391232t^{14} - 238469983891001468288t^{13} - 67911327911152753504t^{12} - 14806387920376690432t^{11} - 2500444267119357232t^{10} - 328511210142795640t^{9} - 33413946732070304t^{8} - 2554414870408092t^{7} - 133456904940280t^{6} - 3498868481426t^{5} + 38046087234t^{4} + 5620560939t^{3} + 238903969t^{2} + 1140681t + 3339 \right)D^{4} + t \left( 644378587334820495360t^{24} + 9910517278425785303040t^{23} + 42383840037427787857920t^{22} + 93591234037303161126912t^{21} + 127349971840028246802432t^{20} + 112743569502095313174528t^{19} + 61680632645815585046528t^{18} + 12273018160695934406656t^{17} - 12028180500730944583680t^{16} - 14507630842073559600128t^{15} - 8789661181747362702336t^{14} - 3685406205296545064704t^{13} - 1155932250135053922304t^{12} - 279468498010503500224t^{11} - 52808537634122123008t^{10} - 7853568041467666912t^{9} - 920684955891966216t^{8} - 84282339843469008t^{7} - 5818895802034668t^{6} - 279426584606120t^{5} - 8046884109946t^{4} - 118013887456t^{3} - 1424909911t^{2} - 11874430t - 5671 \right)D^{3} + 2 t ^{2} \left( 711884915531801690112t^{23} + 10967105936233498411008t^{22} + 45619018189991017709568t^{21} + 97121415183742677024768t^{20} + 126166720727289731825664t^{19} + 104572455148520944902144t^{18} + 50270706980639082883072t^{17} + 3216970392226169271296t^{16} - 16456447082474512158208t^{15} - 15577495113727209548544t^{14} - 8578901069951474897792t^{13} - 3358398735861880828480t^{12} - 991379522026702409824t^{11} - 226223541319951505456t^{10} - 40413255416474692144t^{9} - 5691291637700277920t^{8} - 631903183254384784t^{7} - 54446138604696536t^{6} - 3469625677674592t^{5} - 148235493928698t^{4} - 3644956694356t^{3} - 51422240126t^{2} - 790981728t - 4677561 \right)D^{2} + 8 t ^{2} \left( 193313576200446148608t^{23} + 2981786958847763546112t^{22} + 12142140298469275336704t^{21} + 25131096122995987513344t^{20} + 31473683481543534895104t^{19} + 24698894720398049206272t^{18} + 10493721445374083580928t^{17} - 823883679126980120576t^{16} - 4887362089377894140416t^{15} - 4067908587757799078400t^{14} - 2101551760557551988800t^{13} - 782665270728713847104t^{12} - 220702604527528709088t^{11} - 48173840196098970240t^{10} - 8239209730839826552t^{9} - 1111745769206530276t^{8} - 118114693120622432t^{7} - 9658367107988802t^{6} - 572196538592536t^{5} - 21935202041068t^{4} - 467624007517t^{3} - 6541027311t^{2} - 97690015t - 454104 \right)D + 32 t ^{2} \left( 19725875122494504960t^{23} + 304515630649772605440t^{22} + 1221798621192022130688t^{21} + 2479055298157522944000t^{20} + 3023884514132237279232t^{19} + 2276957183702154006528t^{18} + 869319389025646666752t^{17} - 186867608735591003136t^{16} - 519530345884580774400t^{15} - 401372057522766889600t^{14} - 198781255627167051712t^{13} - 71495633761622258176t^{12} - 19507079663907958624t^{11} - 4121257779527770840t^{10} - 682391173032422992t^{9} - 89147513327564288t^{8} - 9150046062638524t^{7} - 716703225631566t^{6} - 39908464938140t^{5} - 1395028128002t^{4} - 26601570045t^{3} - 381433119t^{2} - 4757227t - 16695 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 4, 6)

|λ| = 17.23689 λ ≈ 17.23689; multiplicity 1; Jordan blocks 1 λ ≈ -4.29494 - 6.469773i; multiplicity 1; Jordan blocks 1 λ ≈ -4.29494 + 6.469773i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 λ ≈ -2 - 2i; multiplicity 1; Jordan blocks 1 λ ≈ -2 + 2i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 3; Jordan blocks 1, 2 λ ≈ -1.647007; multiplicity 1; Jordan blocks 1 3
Sector dimension
10
Spectral radius
$17.2368871$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$17.23689$11
$-4.29494 - 6.469773\mathrm{i}$11
$-4.29494 + 6.469773\mathrm{i}$11
$-3$11
$-2 - 2\mathrm{i}$11
$-2 + 2\mathrm{i}$11
$-2$31, 2
$-1.647007$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda + 3\right) \left(\lambda^{2} + 4 \lambda + 8\right) \left(\lambda^{4} - 7 \lambda^{3} - 102 \lambda^{2} - 1184 \lambda - 1712\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 3\right) \left(\lambda^{2} + 4 \lambda + 8\right) \left(\lambda^{4} - 7 \lambda^{3} - 102 \lambda^{2} - 1184 \lambda - 1712\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 12 & 84 & 36 & 78 & 42 & 336 & 192 & 110 \\ 1 & -1 & 4 & 4 & 0 & 10 & 6 & 54 & 30 & 24 \\ 1 & 1 & 0 & 18 & 10 & 12 & 6 & 84 & 48 & 24 \\ 0 & 1 & 1 & -1 & -1 & 3 & 2 & 0 & 0 & 3 \\ 0 & 1 & 0 & 4 & 0 & 3 & 1 & 10 & 6 & 3 \\ 0 & 0 & 1 & 1 & 1 & 0 & 1 & 18 & 10 & 6 \\ 0 & 0 & 1 & 1 & 1 & -2 & -3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 8 & 10 & -8 & -5 & 20 & 13 & -2 \\ 0 & 0 & 0 & -10 & -16 & 18 & 11 & -32 & -21 & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 14 & 8 & 0\end{pmatrix}\notag\end{equation}$$