Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-113-30-a

·

Blow-up of $\mathbb P^4$ along a canonical curve of genus $5$.

  • $\pi_1\colon X\to\mathbb P^4$ is the blow-up of $\mathbb P^4$ (1-625-126-a) along a canonical curve of genus $5$ and degree $8$, a smooth complete intersection of three quadrics.

  • $\pi_2\colon X\to\mathbb P^2$ is a quartic del Pezzo fibration whose discriminant is a plane curve of degree $20$, the dual of a smooth plane quintic.

Varieties in this family can be blown down to

Defining data
Ambient
ℙ2P2 × ℙ4P4
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w2 + 2w3)
Bundle
Q_1 box O_2(2)
ZL62 label
2040.x351
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 30
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 113
$\chi(X, \mathrm{T}_X)$ = -12

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 113
$\mathrm{c}_1^2\mathrm{c}_2$ = 122
$\mathrm{c}_2^2$ = 100
$\mathrm{c}_1\mathrm{c}_3$ = 34
$\mathrm{c}_4$ = -11
Hodge diamond
1
00
0 2 0
0 5 5 0
0 0 3 0 0
0 5 5 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 + 12 0 x0 + 12 0
$\mathrm{HH}^{ 3 }$ = x1 + 15 0 0 x1 15
$\mathrm{HH}^{ 4 }$ = x1 - 9 0 0 x1 - 42 3 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 25t + 61t^{2} + 25t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(612)
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( 20t^{2} + 8t + 1 \right) ^{2} \left( 50357t^{5} + 11712t^{4} - 152t^{3} - 160t^{2} + 28t - 1 \right) \left( 94235954674359552t^{13} + 124020933383342144t^{12} + 89911469183873440t^{11} + 47543368790142072t^{10} + 17298978891834432t^{9} + 4014187646843232t^{8} + 573019269897776t^{7} + 50113984005084t^{6} + 2913228376636t^{5} + 140358630674t^{4} + 4893367596t^{3} + 84068402t^{2} - 3606897t + 1582 \right)D^{6} + \left( 20t^{2} + 8t + 1 \right) \left( 1993084787205424063226880t^{20} + 3616722609870875528654336t^{19} + 3404585857489905587155584t^{18} + 2238877890447821442319232t^{17} + 1083222901329266384944096t^{16} + 378383673273041457022912t^{15} + 93388624176029700012672t^{14} + 16046436473437101993216t^{13} + 1896827622283262426368t^{12} + 153038420576145164160t^{11} + 8341911341639529932t^{10} + 277618642225545420t^{9} + 6095462104236768t^{8} + 1394554946442504t^{7} + 210465869451435t^{6} + 13666298815408t^{5} + 695095791172t^{4} + 18313648064t^{3} + 381138372t^{2} - 10808035t + 3164 \right)D^{5} + \left( 332180797867570677204480000t^{22} + 700046482752928985340467200t^{21} + 757372722640661489546174976t^{20} + 565797609036648183276711424t^{19} + 314920107048038799657595392t^{18} + 131327524416553945328128000t^{17} + 40762679821026408891294272t^{16} + 9384921479962199742817768t^{15} + 1601966637579313328265280t^{14} + 203047364796388619333728t^{13} + 19208784176898153556288t^{12} + 1367450442630456214916t^{11} + 71912460673575605044t^{10} + 2188996297845738766t^{9} - 64965270244574292t^{8} - 14573689212346954t^{7} - 1017243393431549t^{6} - 40130594561582t^{5} - 1207765998346t^{4} - 25291355216t^{3} - 291487050t^{2} + 6885466t - 1582 \right)D^{4} + t \left( 1395159351043796844258816000t^{21} + 2822672532032792555383705600t^{20} + 2968566248786597350098762240t^{19} + 2180971309962426281473891584t^{18} + 1192005006943752279916064768t^{17} + 483003121370203349421294720t^{16} + 144063613891385594535528800t^{15} + 31641706962872881945535776t^{14} + 5146000660893091911635712t^{13} + 626203270257340722380736t^{12} + 58099886921739086952688t^{11} + 4224737340039275148864t^{10} + 244560559150153034324t^{9} + 10761633267684532428t^{8} + 295912684343617620t^{7} + 325444126543912t^{6} - 357466739827787t^{5} - 11462687636180t^{4} - 246310796728t^{3} - 8490052528t^{2} + 58654540t + 5600 \right)D^{3} + 2 t ^{2} \left( 1541318902105527942228787200t^{20} + 3019046423691047585039567360t^{19} + 3107025770740428683081708032t^{18} + 2254588738793881569034034496t^{17} + 1213364036369560758825402624t^{16} + 479288909284945541670730368t^{15} + 138030744375528594157287504t^{14} + 29100386273916557326690544t^{13} + 4540328569416616233208608t^{12} + 533748843342438028787680t^{11} + 48620544128716195307992t^{10} + 3543378906009210339160t^{9} + 207479714605951761636t^{8} + 9081415891485404048t^{7} + 232949783894382256t^{6} - 877332069103076t^{5} - 294600541591609t^{4} - 7619462207640t^{3} - 75495171772t^{2} - 1881784224t + 12029528 \right)D^{2} + 16 t ^{2} \left( 209273902656569526638822400t^{20} + 400029889216664216224493440t^{19} + 405174631139154900633727232t^{18} + 291397359968633832638853936t^{17} + 154898850163717733325352384t^{16} + 59920399241001157146117696t^{15} + 16769753095078245825678932t^{14} + 3420414120071956454388312t^{13} + 516372511960455025852392t^{12} + 59124718910520964582504t^{11} + 5315375236145135954014t^{10} + 387172622873495853848t^{9} + 22622128970583177182t^{8} + 966075869861528920t^{7} + 22489524264728259t^{6} - 233290843065740t^{5} - 32335988609210t^{4} - 722127328232t^{3} - 4716419816t^{2} - 29219120t + 243628 \right)D + 32 t ^{2} \left( 42708959725830515640576000t^{20} + 80276012925368518350009600t^{19} + 80428846972519333753623552t^{18} + 57495948169977987216878880t^{17} + 30291395036135920360031360t^{16} + 11538657733759818318322240t^{15} + 3161775554352963510757912t^{14} + 629326634793024805085376t^{13} + 92745699496964441337152t^{12} + 10422358693504724323824t^{11} + 928667711981359174836t^{10} + 67535284008044628440t^{9} + 3920692315664799768t^{8} + 163442106764994728t^{7} + 3544450611007290t^{6} - 50141995301184t^{5} - 5233399983773t^{4} - 101478683064t^{3} - 625645105t^{2} + 4043116t + 15820 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ4P4

|λ| = 21.67258 λ ≈ 21.67258; multiplicity 1; Jordan blocks 1 λ ≈ 7.14814 - 8.113245i; multiplicity 1; Jordan blocks 1 λ ≈ 7.14814 + 8.113245i; multiplicity 1; Jordan blocks 1 λ ≈ -4 - 2i; multiplicity 2; Jordan blocks 2 λ ≈ -4 + 2i; multiplicity 2; Jordan blocks 2 λ ≈ -3.984428 - 1.999272i; multiplicity 1; Jordan blocks 1 λ ≈ -3.984428 + 1.999272i; multiplicity 1; Jordan blocks 1 2 2
Sector dimension
9
Spectral radius
$21.6725752$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$21.67258$11
$7.14814 - 8.113245\mathrm{i}$11
$7.14814 + 8.113245\mathrm{i}$11
$-4 - 2\mathrm{i}$22
$-4 + 2\mathrm{i}$22
$-3.984428 - 1.999272\mathrm{i}$11
$-3.984428 + 1.999272\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda^{2} + 8 \lambda + 20\right)^{2} \left(\lambda^{5} - 28 \lambda^{4} + 160 \lambda^{3} + 152 \lambda^{2} - 11712 \lambda - 50357\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda^{2} + 8 \lambda + 20\right)^{2} \left(\lambda^{5} - 28 \lambda^{4} + 160 \lambda^{3} + 152 \lambda^{2} - 11712 \lambda - 50357\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 40 & 24 & 24 & 216 & 320 & 320 & 970 \\ 2 & 0 & 6 & 40 & -2 & 60 & 198 & 12 & 80 \\ 1 & 0 & 4 & 0 & 4 & 40 & 36 & 48 & 160 \\ 0 & 1 & 2 & 4 & 0 & 8 & 40 & -2 & 6 \\ 0 & 2 & 0 & 6 & 0 & 7 & 60 & 40 & 102 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 4 & 12 \\ 0 & 0 & 0 & 1 & 0 & \frac{5}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 1 & 0 & 8 & 4 & 20 \\ 0 & 0 & 0 & 0 & 0 & 0 & 5 & 2 & 0\end{pmatrix}\notag\end{equation}$$