Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-101-27-b

·

Blow-up of the degree-$5$ del Pezzo fourfold along a determinantal surface.

  • $\pi_1\colon X\to W$ is the blow-up of the del Pezzo fourfold $W$ of degree $5$, a codimension-two linear section of $\mathrm{Gr}(2,5)$ (1-405-85-a), along a smooth rational surface of degree $12$ and sectional genus $6$, the blow-up of $\mathbb P^2$ in $12$ points.

  • $\pi_2\colon X\to\mathbb P^3$ is a non-flat conic bundle, given on $W$ (1-405-85-a) by the four quadrics through the surface; its discriminant is a quartic surface with $8$ nodes, over which the fibers are double lines, and over five points the fiber is a cubic scroll $\cong\mathbb F_1$.

Varieties in this family can be blown down to

Defining data
Ambient
ℙ3P3 × 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_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
2 copies of E^(w5) + E^(w1 + w6)
Bundle
2 copies of O_2(1) + O_1(1) box Q_2^v(1)
ZL62 label
3041.44w4
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 27
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 101
$\chi(X, \mathrm{T}_X)$ = -16

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 101
$\mathrm{c}_1^2\mathrm{c}_2$ = 110
$\mathrm{c}_2^2$ = 124
$\mathrm{c}_1\mathrm{c}_3$ = 30
$\mathrm{c}_4$ = 21
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 15 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 + 16 0 x0 + 16 x1
$\mathrm{HH}^{ 3 }$ = x1 + 23 0 0 x1 + 17 6
$\mathrm{HH}^{ 4 }$ = 27 0 0 0 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 22t + 55t^{2} + 22t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

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( 2t + 1 \right) \left( 3t + 1 \right) \left( 128977t^{8} + 245522t^{7} + 200813t^{6} + 92141t^{5} + 25112t^{4} + 3815t^{3} + 251t^{2} - 2t - 1 \right) \left( 132367105205456916t^{17} + 479841313217965300t^{16} + 853195752785329769t^{15} + 998996375863581881t^{14} + 855135810362546727t^{13} + 553114545421720197t^{12} + 270926693724921803t^{11} + 99608603378670840t^{10} + 27142002172513757t^{9} + 5398371572570873t^{8} + 770725044983574t^{7} + 78225078177212t^{6} + 5826599688016t^{5} + 366829398774t^{4} + 22236742170t^{3} + 925484279t^{2} + 3160660t + 39188 \right)D^{6} + \left( 2151111328138611298521432t^{27} + 12935329327369669480293924t^{26} + 37887197997084897797078214t^{25} + 72243226825757489813472167t^{24} + 100829574108900839988294365t^{23} + 109392526638674737783826446t^{22} + 95341690498636660400680317t^{21} + 67953965814056999240026247t^{20} + 39973511751254133879164602t^{19} + 19488635490457023775610745t^{18} + 7884400603954589683292539t^{17} + 2644760600752594656122195t^{16} + 733870067976423563748197t^{15} + 167862255119681783808372t^{14} + 31528763319947001230065t^{13} + 4851192639747520637307t^{12} + 613531098802119747010t^{11} + 64852750572230047073t^{10} + 5941939626559332987t^{9} + 491357370962276630t^{8} + 36423670847613252t^{7} + 2228968433673856t^{6} + 101094678022138t^{5} + 3395537016590t^{4} + 116879615390t^{3} + 3735067576t^{2} + 8933348t + 78376 \right)D^{5} + \left( 17925927734488427487678600t^{27} + 102793796675185219666224780t^{26} + 287888402444068353288827942t^{25} + 526668173690676398681675003t^{24} + 707278730219317387724298342t^{23} + 739179812185477651034205211t^{22} + 619761470014986740330550165t^{21} + 423339381348545842191912386t^{20} + 237304834196762920890889946t^{19} + 109492759126804879662902166t^{18} + 41609647173519113849398751t^{17} + 13011347784475945829278112t^{16} + 3340821831456263084562051t^{15} + 702355694533287481085204t^{14} + 120592229342792006833389t^{13} + 16908730488098446855485t^{12} + 1949202776096576749519t^{11} + 188306724374494417873t^{10} + 15712981707447548731t^{9} + 1149653238657786831t^{8} + 70225634736810851t^{7} + 3085417383432847t^{6} + 67106158327114t^{5} - 908847586069t^{4} - 100580833053t^{3} - 3670599921t^{2} - 10921836t - 39188 \right)D^{4} + t \left( 75288896484851395448250120t^{26} + 415231918561223236788602940t^{25} + 1121683692746426297625666810t^{24} + 1986016867798135191182966601t^{23} + 2588171583517414840281720847t^{22} + 2626808068103908979355364146t^{21} + 2135272131182377837023864825t^{20} + 1408762528894231949194687315t^{19} + 758849855122063361818461348t^{18} + 334531858609099656893150711t^{17} + 120759736355097429693210585t^{16} + 35676779186893200760566973t^{15} + 8617422534923768423430601t^{14} + 1700765559182384445730526t^{13} + 274832425130599338041775t^{12} + 36686332028582784370493t^{11} + 4134242790364242656350t^{10} + 407575751842851529117t^{9} + 36170302123182218179t^{8} + 2829789337743186684t^{7} + 178722065383393124t^{6} + 7947181077667000t^{5} + 206285041186522t^{4} + 2506193140238t^{3} + 29893335642t^{2} + 283218504t + 93896 \right)D^{3} + 2 t ^{2} \left( 83176304688026303542828704t^{25} + 444784262997477781922677764t^{24} + 1168018916167252595369758126t^{23} + 2016442513908368703388874519t^{22} + 2567818956816554208170644063t^{21} + 2547446811596433752937926229t^{20} + 2020459355680275823721931394t^{19} + 1296119882097061472500559984t^{18} + 675852672666406227410235660t^{17} + 287054968628642942392640543t^{16} + 99380957417402506855138211t^{15} + 28050187944500847677994942t^{14} + 6456976676347166463546306t^{13} + 1214890068029433229925662t^{12} + 188180789822131936144355t^{11} + 24410936225736173605129t^{10} + 2733866757574028307987t^{9} + 273345818897301601078t^{8} + 24497668551301316032t^{7} + 1865517188702393824t^{6} + 108959812116384350t^{5} + 4257181399077772t^{4} + 92920240774502t^{3} + 1080087384214t^{2} + 17356209190t + 101945764 \right)D^{2} + 4 t ^{2} \left( 45173337890910837268950072t^{25} + 236012538372883623815572224t^{24} + 606783675847712989585105104t^{23} + 1028008501814430330610434256t^{22} + 1286777832415611515349548838t^{21} + 1254812369340049839997598608t^{20} + 976585007289963855828817342t^{19} + 612894560094076773932790066t^{18} + 311514291606966965564223166t^{17} + 128474122751979905784106789t^{16} + 43036206526455856973800513t^{15} + 11720003369655038984510927t^{14} + 2599775867788496204375072t^{13} + 472182839499131273361658t^{12} + 71094213649549694734232t^{11} + 9091166051473514618227t^{10} + 1021729397262131991903t^{9} + 103339284610693045764t^{8} + 9221041459913727570t^{7} + 675201625768364796t^{6} + 36448461159050997t^{5} + 1264017777751761t^{4} + 23751733275067t^{3} + 280684508604t^{2} + 4363572540t + 18183232 \right)D + 8 t ^{2} \left( 9219048549165476993663280t^{25} + 47400423996573898839470160t^{24} + 120095467227289322244871416t^{23} + 200836631545519485770396016t^{22} + 248407546863130901092948990t^{21} + 239327389891232451434641023t^{20} + 183760850979398259726383223t^{19} + 113508277023003651762511543t^{18} + 56621084801055697852763962t^{17} + 22850106952407127912982024t^{16} + 7469481996575198356006740t^{15} + 1980914523692647570106239t^{14} + 427633564301554489394215t^{13} + 75747420822375674573140t^{12} + 11198360029309979807038t^{11} + 1423136572199546624306t^{10} + 160922819333900684069t^{9} + 16385858752946800051t^{8} + 1446713548891235321t^{7} + 101958264228322986t^{6} + 5140340012367316t^{5} + 161219376445462t^{4} + 2701997661978t^{3} + 34150197706t^{2} + 434960804t + 1254016 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × Gr(2, 5)

|λ| = 21.29559 λ ≈ 21.29559; multiplicity 1; Jordan blocks 1 λ ≈ -6.216821 - 4.542907i; multiplicity 1; Jordan blocks 1 λ ≈ -6.216821 + 4.542907i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 λ ≈ -2.852474; multiplicity 1; Jordan blocks 1 λ ≈ -1.73572 - 2.028795i; multiplicity 1; Jordan blocks 1 λ ≈ -1.73572 + 2.028795i; multiplicity 1; Jordan blocks 1 λ ≈ -2.622089; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ -1.915943; multiplicity 1; Jordan blocks 1
Sector dimension
10
Spectral radius
$21.2955878$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$21.29559$11
$-6.216821 - 4.542907\mathrm{i}$11
$-6.216821 + 4.542907\mathrm{i}$11
$-3$11
$-2.852474$11
$-1.73572 - 2.028795\mathrm{i}$11
$-1.73572 + 2.028795\mathrm{i}$11
$-2.622089$11
$-2$11
$-1.915943$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right) \left(\lambda^{8} + 2 \lambda^{7} - 251 \lambda^{6} - 3815 \lambda^{5} - 25112 \lambda^{4} - 92141 \lambda^{3} - 200813 \lambda^{2} - 245522 \lambda - 128977\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right) \left(\lambda^{8} + 2 \lambda^{7} - 251 \lambda^{6} - 3815 \lambda^{5} - 25112 \lambda^{4} - 92141 \lambda^{3} - 200813 \lambda^{2} - 245522 \lambda - 128977\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 14 & 18 & 138 & 63 & 63 & 102 & 392 & 616 & 1240 \\ 1 & -1 & 4 & 16 & 10 & 8 & 14 & 75 & 117 & 224 \\ 1 & 2 & 0 & 28 & 14 & 10 & 16 & 72 & 114 & 224 \\ 0 & 1 & 1 & -1 & 0 & 2 & 3 & 8 & 12 & 36 \\ 0 & 1 & 0 & 4 & -1 & 1 & 2 & 6 & 10 & 24 \\ 0 & 0 & 1 & 2 & 1 & -3 & -2 & 0 & 0 & 0 \\ 0 & 0 & 1 & 2 & 2 & 1 & 0 & 14 & 22 & 36 \\ 0 & 0 & 0 & 15 & 19 & -6 & -\frac{21}{2} & 15 & 29 & 12 \\ 0 & 0 & 0 & -7 & -11 & 5 & \frac{17}{2} & -8 & -16 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{4} & \frac{11}{4} & 0\end{pmatrix}\notag\end{equation}$$