Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-177-42-a

·

Blow-up of a smooth quadric fourfold along a smooth surface isomorphic to $\mathbb P^2$ blown up at nine points.

  • $\pi_1\colon X\to Y$ maps birationally onto a fourfold in $\mathrm{Gr}(3,6)$ with a unique singular point; it contracts a plane with normal bundle $\mathcal O(-1)^{\oplus 2}$ to that point and is an isomorphism elsewhere.

  • $\pi_2\colon X\to Q^4$ is the blow-up of the smooth quadric fourfold (1-512-105-a) along a smooth surface of degree $7$ and sectional genus $3$, the blow-up of $\mathbb P^2$ in $9$ points embedded by quartics through them.

Varieties in this family can be blown down to

Defining data
Ambient
Fl(2, 3, 6)
node 1: fundamental weight ω_1node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
2 copies of E^(w1) + 3 copies of E^(w3)
Bundle
2 copies of U_3^v + 3 copies of O(0,1)
ZL62 label
55.22200
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,4) \\ \det\mathcal{E} &\cong \mathcal{O}(2,3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 42
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 177
$\chi(X, \mathrm{T}_X)$ = -12

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 177
$\mathrm{c}_1^2\mathrm{c}_2$ = 138
$\mathrm{c}_2^2$ = 109
$\mathrm{c}_1\mathrm{c}_3$ = 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 + 12 0 x0 + 12 x1
$\mathrm{HH}^{ 3 }$ = x1 + 37 0 0 x1 + 3 34
$\mathrm{HH}^{ 4 }$ = 42 0 0 0 0 42
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 37t + 101t^{2} + 37t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(350)
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( 1139t^{9} + 220t^{8} + 689t^{7} + 1537t^{6} + 289t^{5} + 911t^{4} + 1071t^{3} + 88t^{2} - 2t - 1 \right) \left( 1792906607896t^{17} + 385841953752t^{16} - 7496264461040t^{15} + 4440138695568t^{14} - 15646326859663t^{13} - 12363904199536t^{12} + 10324389663173t^{11} + 5375974981217t^{10} - 8087694631155t^{9} + 4662725687472t^{8} + 1657847066109t^{7} + 725756072429t^{6} - 470793383313t^{5} + 26164846356t^{4} - 1479728733t^{3} + 1867454397t^{2} + 19442340t + 237888 \right)D^{6} + \left( 42884533154264424t^{27} + 51089949039338904t^{26} - 165754396827676424t^{25} - 31356097580430576t^{24} - 402099674289036085t^{23} - 925272563242883797t^{22} - 280879610915086078t^{21} - 139111900184384733t^{20} - 524138726284641240t^{19} + 50559365086648685t^{18} + 332986850605066138t^{17} + 81049162412331118t^{16} + 190530973430048346t^{15} + 136848967589610785t^{14} + 38786618120342275t^{13} + 66044001041936599t^{12} + 33322612345743632t^{11} + 9852866742447296t^{10} - 4692495540808939t^{9} - 2095512667845360t^{8} + 169445189619897t^{7} + 38885938043002t^{6} + 14300792264055t^{5} + 1004358169269t^{4} + 3718684455t^{3} + 7534240776t^{2} + 56899692t + 475776 \right)D^{5} + \left( 357371109618870200t^{27} + 363704466426496680t^{26} - 1567937291053744480t^{25} - 123151027094524760t^{24} - 3897980804397176771t^{23} - 7576355539225441999t^{22} - 974082814962776498t^{21} + 1411693995906355318t^{20} - 2633738120180331879t^{19} + 1143003024564398500t^{18} + 3278950099286114412t^{17} + 861316062976777037t^{16} + 1027066525977245009t^{15} + 790746723377118158t^{14} + 577029788779703146t^{13} + 245601887907570208t^{12} - 5249910192389040t^{11} - 14447438706936992t^{10} - 26654270566546288t^{9} - 2093657653879478t^{8} + 534573860051162t^{7} + 157603496015077t^{6} + 63276886387824t^{5} + 1607003575491t^{4} + 48207207750t^{3} - 7161533847t^{2} - 59713992t - 237888 \right)D^{4} + t \left( 1500958660399254840t^{26} + 1322809899616905240t^{25} - 7139783981527822040t^{24} + 388268248665731152t^{23} - 18592988523318973903t^{22} - 32090579783295902143t^{21} + 1113268219287614546t^{20} + 13141971289125435561t^{19} - 9549108560670354872t^{18} + 6692601288684779891t^{17} + 13630236064417532574t^{16} + 4524594030514693826t^{15} + 4311814017028907204t^{14} + 3116784785097194795t^{13} + 2106352708995183283t^{12} + 75599198647848509t^{11} - 228429906333276244t^{10} - 109498989569934048t^{9} - 53824616160424559t^{8} + 5305296385678616t^{7} + 684341721081415t^{6} + 428946248699982t^{5} + 96827504590647t^{4} + 2248711280571t^{3} + 46108747863t^{2} + 458913060t + 133056 \right)D^{3} + 2 t ^{2} \left( 1658201948631557728t^{25} + 1288328448438902004t^{24} - 8318271511797680936t^{23} + 1411817123525610808t^{22} - 22860604261882273676t^{21} - 35733136918525387428t^{20} + 7417282706372829675t^{19} + 20702585102995131955t^{18} - 10782009789273800063t^{17} + 10394046636091705709t^{16} + 16534601942456586515t^{15} + 7612744899927526713t^{14} + 5182119026997123825t^{13} + 2944985577296680918t^{12} + 1512356592415951383t^{11} - 423837933315233023t^{10} - 253619213262695514t^{9} - 99335492649216586t^{8} - 21592636833640376t^{7} + 6198848188073746t^{6} + 174370173251805t^{5} + 305583669173907t^{4} + 45773031496692t^{3} + 969990288534t^{2} + 23980920129t + 178967178 \right)D^{2} + 4 t ^{2} \left( 900575196239552904t^{25} + 630817528904158404t^{24} - 4678503674975402864t^{23} + 1207032906282417392t^{22} - 13482209168862428689t^{21} - 19413056788921129513t^{20} + 7207743275416453439t^{19} + 13656845581421187226t^{18} - 6413983454061693461t^{17} + 7478138215569348988t^{16} + 10104098573920988161t^{15} + 5462973601516898771t^{14} + 2779872658879811339t^{13} + 1259883952205718237t^{12} + 441037265271891404t^{11} - 332269751980034570t^{10} - 116429081173191764t^{9} - 39586029158307504t^{8} - 1701881247666657t^{7} + 2355614601368328t^{6} + 26730873930366t^{5} + 113346916511481t^{4} + 11571053511501t^{3} + 265297961043t^{2} + 6440074560t + 35683200 \right)D + 48 t ^{2} \left( 30631809395903160t^{25} + 19873595143815060t^{24} - 162681013118929280t^{23} + 51778102525863284t^{22} - 485213899048960543t^{21} - 657923119116303255t^{20} + 329691720820855747t^{19} + 517880443890189660t^{18} - 238357607000874612t^{17} + 307551841496387865t^{16} + 376342686026669840t^{15} + 217653575736294195t^{14} + 88694719802200566t^{13} + 33266205439444651t^{12} + 6333304028709625t^{11} - 12301876002992040t^{10} - 3304529077006256t^{9} - 1001675930855203t^{8} + 97692673708690t^{7} + 52374516448288t^{6} + 831850047462t^{5} + 2821646289711t^{4} + 206107348434t^{3} + 5589887751t^{2} + 120385440t + 475776 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 3, 6)

|λ| = 12.45618 λ ≈ 12.45618; multiplicity 1; Jordan blocks 1 λ ≈ -6.774793 - 5.807629i; multiplicity 1; Jordan blocks 1 λ ≈ -6.774793 + 5.807629i; multiplicity 1; Jordan blocks 1 λ ≈ -1.160161 - 0.2168122i; multiplicity 1; Jordan blocks 1 λ ≈ -1.160161 + 0.2168122i; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 1; Jordan blocks 1 λ ≈ 0.09367886 - 0.9505175i; multiplicity 1; Jordan blocks 1 λ ≈ 0.09367886 + 0.9505175i; multiplicity 1; Jordan blocks 1 λ ≈ 0.6131868 - 0.7264224i; multiplicity 1; Jordan blocks 1 λ ≈ 0.6131868 + 0.7264224i; multiplicity 1; Jordan blocks 1
Sector dimension
10
Spectral radius
$12.4561769$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$12.45618$11
$-6.774793 - 5.807629\mathrm{i}$11
$-6.774793 + 5.807629\mathrm{i}$11
$-1.160161 - 0.2168122\mathrm{i}$11
$-1.160161 + 0.2168122\mathrm{i}$11
$-1$11
$0.09367886 - 0.9505175\mathrm{i}$11
$0.09367886 + 0.9505175\mathrm{i}$11
$0.6131868 - 0.7264224\mathrm{i}$11
$0.6131868 + 0.7264224\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda^{9} + 2 \lambda^{8} - 88 \lambda^{7} - 1071 \lambda^{6} - 911 \lambda^{5} - 289 \lambda^{4} - 1537 \lambda^{3} - 689 \lambda^{2} - 220 \lambda - 1139\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda^{9} + 2 \lambda^{8} - 88 \lambda^{7} - 1071 \lambda^{6} - 911 \lambda^{5} - 289 \lambda^{4} - 1537 \lambda^{3} - 689 \lambda^{2} - 220 \lambda - 1139\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 6 & 6 & 12 & 33 & 9 & 33 & 12 & 12 & 50 \\ 1 & 0 & 3 & 0 & -6 & 0 & -6 & 15 & 12 & 0 \\ 1 & 0 & -1 & 6 & 12 & 4 & 10 & 6 & 3 & 8 \\ 0 & 1 & 0 & 1 & 3 & 0 & 3 & 0 & 0 & 6 \\ 0 & 1 & 2 & -1 & -1 & 0 & 0 & 2 & 4 & 6 \\ 0 & 2 & 1 & -1 & 2 & 0 & 3 & -4 & -2 & 6 \\ 0 & 0 & 1 & 1 & 0 & 0 & -1 & 4 & 2 & 0 \\ 0 & 0 & 0 & 1 & \frac{20}{3} & \frac{4}{3} & 8 & -1 & 0 & 8 \\ 0 & 0 & 0 & 2 & -\frac{1}{3} & \frac{4}{3} & -2 & 2 & 0 & -4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 2 & 0\end{pmatrix}\notag\end{equation}$$