Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-118-30-a

·

Blow-up of a smooth quadric fourfold along a determinantal surface.

  • $\pi_1\colon X\to Q$ is the blow-up of the smooth quadric fourfold $Q=\mathrm{Gr}(2,4)$ (1-512-105-a) along a smooth surface $S\subset Q$ of degree $9$ and sectional genus $5$, isomorphic to the blow-up of $\mathbb P^2$ at ten distinct points and embedded by septics with double points there.

  • $\pi_2\colon X\to Z\subset\mathbb P^6$ maps birationally onto the determinantal sextic fourfold $Z$ cut out by the $3\times3$ minors of a general $3\times4$ matrix of linear forms; it contracts ten disjoint planes with normal bundle $\mathcal O_{\mathbb P^2}(-1)^{\oplus 2}$ to the ten singular points of $Z$, each locally the cone over $\mathbb P^2\times\mathbb P^1$.

Varieties in this family can be blown down to

Defining data
Ambient
ℙ6P6 × Gr(2, 4)
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_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8 (marked)node 9: fundamental weight ω_9ω1ω2ω3ω4ω5ω6ω7ω8ω9
Bundle
3 copies of E^(w1 + w7)
Bundle
3 copies of O_1(1) box U_2^v
ZL62 label
31600.w2w2w2
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(7;4) \\ \det\mathcal{E} &\cong \mathcal{O}(6;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 30
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 118
$\chi(X, \mathrm{T}_X)$ = -12

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 118
$\mathrm{c}_1^2\mathrm{c}_2$ = 112
$\mathrm{c}_2^2$ = 125
$\mathrm{c}_1\mathrm{c}_3$ = 34
$\mathrm{c}_4$ = 19
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 13 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 + 24 0 0 x1 + 12 12
$\mathrm{HH}^{ 4 }$ = 30 0 0 0 0 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 25t + 66t^{2} + 25t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

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( 344t^{9} + 3404t^{8} + 9756t^{7} + 455t^{6} + 17514t^{5} + 3818t^{4} + 1841t^{3} + 199t^{2} - 3t - 1 \right) \left( 153148408335360t^{17} + 1082374503137280t^{16} + 2363438127278080t^{15} + 1421913152107520t^{14} + 314262351483488t^{13} - 739894874078336t^{12} - 921910876474000t^{11} - 431819577858496t^{10} - 125094056864876t^{9} - 86817415395908t^{8} - 43781116637020t^{7} - 16356505536784t^{6} - 1640661767530t^{5} + 759314490t^{4} - 15441006451t^{3} - 899719327t^{2} - 6748616t - 73396 \right)D^{6} + \left( 895611891945185280t^{26} + 13479186289082388480t^{25} + 80476113766877696000t^{24} + 236404782420354483200t^{23} + 338985158827625389312t^{22} + 222739228554353333376t^{21} + 106345162647699922432t^{20} - 84662717046826514080t^{19} - 165135572514959708320t^{18} - 172944967165250542752t^{17} - 152575022799553988176t^{16} - 88255800153939482036t^{15} - 33720128887669546864t^{14} - 20323329228524499500t^{13} - 10800545534208114452t^{12} - 4888993863413461414t^{11} - 1091745232086088124t^{10} - 171210669882612851t^{9} - 20664539579659660t^{8} - 1809109858036520t^{7} - 387535637979459t^{6} - 34307522627409t^{5} - 873684698201t^{4} - 85271125535t^{3} - 3666266284t^{2} - 19805472t - 146792 \right)D^{5} + \left( 6058551033746841600t^{26} + 82425760695159828480t^{25} + 444076535039752540160t^{24} + 1182339262360454241280t^{23} + 1565728668359413553920t^{22} + 849481921766043844480t^{21} - 87276788646538636800t^{20} - 841552789146710498016t^{19} - 826186227522518594784t^{18} - 496940346509680898304t^{17} - 392345578131989262912t^{16} - 283227825348173179116t^{15} - 144139502378328864612t^{14} - 90897577488809165620t^{13} - 43595933917399779156t^{12} - 18616020786142993966t^{11} - 3273062282551171578t^{10} - 227888899504336559t^{9} - 27398424480323669t^{8} - 1647523866846128t^{7} - 320742767024273t^{6} - 24110433749460t^{5} - 2024942293658t^{4} + 44134818968t^{3} + 3170410539t^{2} + 19358960t + 73396 \right)D^{4} + t \left( 20809805724608716800t^{25} + 260889402029719449600t^{24} + 1286146928716256276480t^{23} + 3105543175935799741440t^{22} + 3780801316708992198400t^{21} + 2089905022052377312640t^{20} - 515799143773731118080t^{19} - 2781471928618860625632t^{18} - 2721182847317148701664t^{17} - 1392651597805744808800t^{16} - 980502570848568479952t^{15} - 812148166431912776252t^{14} - 485847108035502134312t^{13} - 257349636689615059620t^{12} - 95265089733585166004t^{11} - 38019826800737295586t^{10} - 6070108576680104008t^{9} - 132633652865658913t^{8} - 58497221755069558t^{7} - 6716047782987296t^{6} - 773788254542769t^{5} - 50341985425311t^{4} - 1205583059417t^{3} - 24072512765t^{2} - 253629510t - 63444 \right)D^{3} + 2 t ^{2} \left( 19071264993185710080t^{24} + 224414651744923115520t^{23} + 1027182237535857111040t^{22} + 2267085855415819048960t^{21} + 2534524754562442653184t^{20} + 1452217613085377209856t^{19} - 379131152092737329792t^{18} - 2279487480044305181056t^{17} - 2473595530465335567168t^{16} - 1345330402273002075280t^{15} - 894224765771717000960t^{14} - 706076313302708358272t^{13} - 420071010737815307016t^{12} - 196044750440962048232t^{11} - 55535285487674704600t^{10} - 20579975691815884956t^{9} - 3341194175671622200t^{8} + 19132058492554034t^{7} - 37798765129177308t^{6} - 4196626030617706t^{5} - 209690215419653t^{4} - 17891646261780t^{3} - 460808481031t^{2} - 12431094516t - 96798586 \right)D^{2} + 4 t ^{2} \left( 8798069762049761280t^{24} + 98773483605917921280t^{23} + 425980132745985105920t^{22} + 869390247915711738880t^{21} + 901226956256677480192t^{20} + 515005835516234408576t^{19} - 161667646355164492928t^{18} - 945128268200432892448t^{17} - 1116301387402326259232t^{16} - 665345070257071687920t^{15} - 437159981526469000336t^{14} - 318161153912741189260t^{13} - 179583891600307488312t^{12} - 77307929130257508604t^{11} - 17212107248229678612t^{10} - 5692994787524433574t^{9} - 1021262128290670356t^{8} + 19424763793464785t^{7} - 12972095482382904t^{6} - 1330328844536710t^{5} - 19062019626382t^{4} - 3850061831699t^{3} - 113903733766t^{2} - 3163634188t - 18789376 \right)D + 32 t ^{2} \left( 395122893505228800t^{24} + 4291056392075458560t^{23} + 17690969436232028160t^{22} + 33869823973091543040t^{21} + 33085481264628325760t^{20} + 18197496968865691200t^{19} - 7940765150151736960t^{18} - 39255909731652533968t^{17} - 48651386344241810208t^{16} - 31182150264183041488t^{15} - 20595932424602593736t^{14} - 14058789203968907432t^{13} - 7511501162420040956t^{12} - 3068871275259507580t^{11} - 563166925403038786t^{10} - 158692151107353574t^{9} - 33053421132155238t^{8} + 832208953044863t^{7} - 437419661206424t^{6} - 42155594149660t^{5} + 183119889076t^{4} - 97641109875t^{3} - 3391308916t^{2} - 82545620t - 366980 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ6P6 × Gr(2, 4)

|λ| = 16.64953 λ ≈ 16.64953; multiplicity 1; Jordan blocks 1 λ ≈ -9.349571 - 3.516067i; multiplicity 1; Jordan blocks 1 λ ≈ -9.349571 + 3.516067i; multiplicity 1; Jordan blocks 1 λ ≈ -0.5339055 - 3.149485i; multiplicity 1; Jordan blocks 1 λ ≈ -0.5339055 + 3.149485i; multiplicity 1; Jordan blocks 1 λ ≈ 0.2224328 - 0.7696488i; multiplicity 1; Jordan blocks 1 λ ≈ 0.2224328 + 0.7696488i; multiplicity 1; Jordan blocks 1 λ ≈ -0.1637192 - 0.06937439i; multiplicity 1; Jordan blocks 1 λ ≈ -0.1637192 + 0.06937439i; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 1; Jordan blocks 1
Sector dimension
10
Spectral radius
$16.6495253$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$16.64953$11
$-9.349571 - 3.516067\mathrm{i}$11
$-9.349571 + 3.516067\mathrm{i}$11
$-0.5339055 - 3.149485\mathrm{i}$11
$-0.5339055 + 3.149485\mathrm{i}$11
$0.2224328 - 0.7696488\mathrm{i}$11
$0.2224328 + 0.7696488\mathrm{i}$11
$-0.1637192 - 0.06937439\mathrm{i}$11
$-0.1637192 + 0.06937439\mathrm{i}$11
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda^{9} + 3 \lambda^{8} - 199 \lambda^{7} - 1841 \lambda^{6} - 3818 \lambda^{5} - 17514 \lambda^{4} - 455 \lambda^{3} - 9756 \lambda^{2} - 3404 \lambda - 344\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda^{9} + 3 \lambda^{8} - 199 \lambda^{7} - 1841 \lambda^{6} - 3818 \lambda^{5} - 17514 \lambda^{4} - 455 \lambda^{3} - 9756 \lambda^{2} - 3404 \lambda - 344\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 10 & 10 & 60 & 81 & 27 & 12 & 252 & 132 & 330 \\ 1 & -1 & 0 & 24 & 34 & 10 & 4 & 63 & 27 & 120 \\ 1 & 3 & 0 & 4 & -6 & 10 & 4 & 18 & 27 & 12 \\ 0 & 1 & 1 & -1 & 0 & -3 & 0 & 12 & 6 & 0 \\ 0 & 1 & 0 & 0 & -1 & 3 & 0 & 6 & 6 & 9 \\ 0 & 0 & 1 & 3 & 6 & 1 & 0 & 18 & 6 & 27 \\ 0 & 0 & 1 & 3 & 3 & 0 & 0 & -18 & -12 & 9 \\ 0 & 0 & 0 & \frac{2}{3} & -1 & \frac{5}{3} & \frac{2}{3} & 1 & 2 & 0 \\ 0 & 0 & 0 & \frac{13}{3} & \frac{20}{3} & -\frac{1}{3} & \frac{2}{3} & -1 & -2 & 10 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 2 & 0\end{pmatrix}\notag\end{equation}$$