Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-305-66-a

·

Blow-up of $\mathbb P^4$ along an elliptic quartic curve.

  • $\pi_1\colon X\to\mathbb P^4$ is the blow-up of $\mathbb P^4$ (1-625-126-a) along a smooth elliptic quartic curve, the complete intersection of two quadrics in a hyperplane $\mathbb P^3$.

  • $\pi_2\colon X\to Y$ is the blow-up of a complete intersection $Y$ of two quadrics in $\mathbb P^6$ singular along a line (transversal $3$-fold nodes along it), along that line; it contracts the strict transform of the hyperplane $\mathbb P^3$ above onto the line.

Varieties in this family can be blown up to

Varieties in this family can be blown down to

  • 1-625-126-a
  • A complete intersection of two quadrics in $\mathbb P^6$ singular along a line
Defining data
Ambient
Fl(1, 3, 7)
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
2 copies of E^(w6) + E^(w2 + w3)
Bundle
2 copies of wedge^3 Q_1^v ot wedge^2 U_2^v ot U_3^v + Sigma_(2,1) U_2^v ot Sym^2 U_3^v
ZL62 label
604.55l
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,6) \\ \det\mathcal{E} &\cong \mathcal{O}(1,5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 66
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 305
$\chi(X, \mathrm{T}_X)$ = 4

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 305
$\mathrm{c}_1^2\mathrm{c}_2$ = 170
$\mathrm{c}_2^2$ = 100
$\mathrm{c}_1\mathrm{c}_3$ = 50
$\mathrm{c}_4$ = 5
Hodge diamond
1
00
0 2 0
0 1 1 0
0 0 3 0 0
0 1 1 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 + x2 - 4 0 x0 - 4 x2
$\mathrm{HH}^{ 3 }$ = x1 + 84 0 0 x1 84
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 92 0 0 x1 - x2 + 26 0 66
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 61t + 181t^{2} + 61t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

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( 4t^{2} + 1 \right) ^{2} \left( 10037t^{5} + 3600t^{4} + 500t^{3} - 16t^{2} - 8t - 1 \right) \left( 301840270598496t^{11} + 340791108540552t^{10} + 212372845698072t^{9} + 96165469845240t^{8} + 21593061035256t^{7} + 2045183733316t^{6} + 256925243612t^{5} + 88708165882t^{4} + 7415704556t^{3} + 392759890t^{2} + 708683t + 87840 \right)D^{6} + \left( 4t^{2} + 1 \right) \left( 254483946863756765568t^{18} + 374896289607605618112t^{17} + 313493833150234181952t^{16} + 190757970200419834176t^{15} + 79414143392824653336t^{14} + 25024734557049826192t^{13} + 6147360082797225720t^{12} + 1135877914867920848t^{11} + 183339226635009356t^{10} + 37535439992999924t^{9} + 7006687275384648t^{8} + 846876812447992t^{7} + 56424092179799t^{6} + 4042361933284t^{5} + 557891209640t^{4} + 37384872008t^{3} + 1552721136t^{2} + 720609t + 175680 \right)D^{5} + \left( 8482798228791892185600t^{20} + 12397771835965859592960t^{19} + 11611390672493154147072t^{18} + 8170419559507814560320t^{17} + 4091648532450057568512t^{16} + 1685887802439749018328t^{15} + 534185526835340511064t^{14} + 128372738772543135560t^{13} + 25411118233822233368t^{12} + 4446018383227199980t^{11} + 703570013302873156t^{10} + 133547959398606310t^{9} + 19170723493514468t^{8} + 1816947918121918t^{7} + 132273986416423t^{6} + 5683275469572t^{5} - 398552399658t^{4} - 29695292944t^{3} - 1588507540t^{2} - 3767254t - 87840 \right)D^{4} + t \left( 35627752560925947179520t^{19} + 51744696912907328954880t^{18} + 46199236307092346225280t^{17} + 30943887598244736972864t^{16} + 14267666710566034383648t^{15} + 5322982783399130422688t^{14} + 1535657711964096474120t^{13} + 323184968301848468784t^{12} + 56549568160434926680t^{11} + 10578441820096383968t^{10} + 1791943832180238548t^{9} + 341107175711582364t^{8} + 46259956591010332t^{7} + 4040444231474008t^{6} + 253469981792425t^{5} + 8741226814748t^{4} + 91736632860t^{3} + 3616557248t^{2} + 7698892t + 7488 \right)D^{3} + 2 t ^{2} \left( 39360183781594379741184t^{18} + 56890068962490489907008t^{17} + 49237864509656489855040t^{16} + 31917649601191357217664t^{15} + 13807666711008760053168t^{14} + 4723526835258162170192t^{13} + 1267937620562894459288t^{12} + 243957166123314321664t^{11} + 39356776051619870216t^{10} + 8265277408020414248t^{9} + 1428903716406765156t^{8} + 239515450043028584t^{7} + 27704383082229744t^{6} + 2147110891934844t^{5} + 129437773718011t^{4} + 4407108702000t^{3} + 64744219634t^{2} + 2192221820t + 2152385 \right)D^{2} + 16 t ^{2} \left( 5344162884138892076928t^{18} + 7696885969122321804912t^{17} + 6531593014175636762064t^{16} + 4147580595041717709168t^{15} + 1712969404974905090124t^{14} + 545871848555614520376t^{13} + 138736310395944626400t^{12} + 25316434221531304632t^{11} + 3916568433692966220t^{10} + 925149455342760984t^{9} + 160197926399085116t^{8} + 23206566911573750t^{7} + 2272881910682069t^{6} + 156871578448166t^{5} + 8848068558930t^{4} + 302737674340t^{3} + 5906010680t^{2} + 168957856t + 58560 \right)D + 144 t ^{3} \left( 242365663679768348160t^{17} + 348225232576797522240t^{16} + 291983874202361449152t^{15} + 183131973791539725696t^{14} + 73317092555471655056t^{13} + 22173190103908590848t^{12} + 5418687351680256656t^{11} + 960972405026932800t^{10} + 146026303574492248t^{9} + 37743550806499040t^{8} + 6503650137449400t^{7} + 836709145370200t^{6} + 70862414241404t^{5} + 4394143778260t^{4} + 225047472050t^{3} + 7886922924t^{2} + 193125033t + 4743360 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 7)

|λ| = 7.894861 λ ≈ 7.894861; multiplicity 1; Jordan blocks 1 λ ≈ -4.061649 - 6.238423i; multiplicity 1; Jordan blocks 1 λ ≈ -4.061649 + 6.238423i; multiplicity 1; Jordan blocks 1 λ ≈ -3.885781 - 2.800496i; multiplicity 1; Jordan blocks 1 λ ≈ -3.885781 + 2.800496i; multiplicity 1; Jordan blocks 1 λ ≈ 0 - 2i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 0 + 2i; multiplicity 2; Jordan blocks 1, 1 2 2
Sector dimension
9
Spectral radius
$7.89486096$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$7.894861$11
$-4.061649 - 6.238423\mathrm{i}$11
$-4.061649 + 6.238423\mathrm{i}$11
$-3.885781 - 2.800496\mathrm{i}$11
$-3.885781 + 2.800496\mathrm{i}$11
$0 - 2\mathrm{i}$21, 1
$0 + 2\mathrm{i}$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda^{2} + 4\right)^{2} \left(\lambda^{5} + 8 \lambda^{4} + 16 \lambda^{3} - 500 \lambda^{2} - 3600 \lambda - 10037\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda^{2} + 4\right) \left(\lambda^{5} + 8 \lambda^{4} + 16 \lambda^{3} - 500 \lambda^{2} - 3600 \lambda - 10037\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 12 & 12 & 12 & 0 & 0 & 10 \\ 2 & 0 & 2 & 0 & -2 & -4 & 6 & 12 & 0 \\ 1 & 0 & -2 & 0 & 4 & 4 & 12 & 12 & 0 \\ 0 & 1 & 2 & -2 & 0 & -1 & 4 & -2 & 0 \\ 0 & 2 & 0 & 2 & 0 & 3 & -4 & 0 & 6 \\ 0 & 0 & 1 & 0 & 0 & -2 & 0 & 4 & 0 \\ 0 & 0 & 0 & 1 & 0 & \frac{5}{2} & -2 & 0 & 0 \\ 0 & 0 & 0 & 2 & 3 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 5 & 6 & 0\end{pmatrix}\notag\end{equation}$$