Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-129-33-a

·

Blow-up of $\mathbb P^4$ along a minimal elliptic surface of degree $8$.

  • $\pi_1\colon X\to\mathrm{Gr}(2,7)$ maps birationally onto a fourfold of degree $16$; it contracts a plane with normal bundle $\mathcal O(-1)^{\oplus 2}$ to a point and embeds the rest, and the normalization of the image has one singular point, locally the cone over $\mathbb P^2\times\mathbb P^1\subset\mathbb P^5$.

  • $\pi_2\colon X\to\mathbb P^4$ is the blow-up of $\mathbb P^4$ (1-625-126-a) along a smooth minimal elliptic surface of degree $8$ with $p_g=2$ and $q=0$, the rank-one locus of a general $2\times3$ matrix of forms of degrees $1,2,2$.

Varieties in this family can be blown down to

Defining data
Ambient
ℙ4P4 × Gr(2, 7)
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_6 (marked)node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9node 10: fundamental weight ω_10ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10
Bundle
2 copies of E^(w1 + w6) + E^(w4 + w5)
Bundle
2 copies of O_1(1) box O_2(1) + Q_1 box U_2^v
ZL62 label
40601.wOw4w4
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;7) \\ \det\mathcal{E} &\cong \mathcal{O}(4;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 33
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 129
$\chi(X, \mathrm{T}_X)$ = -28

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 129
$\mathrm{c}_1^2\mathrm{c}_2$ = 126
$\mathrm{c}_2^2$ = 124
$\mathrm{c}_1\mathrm{c}_3$ = 14
$\mathrm{c}_4$ = 41
Hodge diamond
1
00
0 2 0
0 0 0 0
0 2 31 2 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 + x2 + 28 0 x0 + 28 x2
$\mathrm{HH}^{ 3 }$ = x1 + 22 0 0 x1 22
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 34 0 0 x1 - x2 + 1 0 33
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 28t + 71t^{2} + 28t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(441)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(498)

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) ^{4} \left( 1973t^{5} - 1144t^{4} + 1922t^{3} + 128t^{2} + t - 1 \right) \left( 35958275020t^{9} + 120563862792t^{8} + 164036146407t^{7} + 116431933137t^{6} + 45640993829t^{5} + 5559987535t^{4} - 994202811t^{3} + 526991795t^{2} + 4076303t + 41925 \right)D^{6} + \left( t + 1 \right) ^{3} \left( 1489859208903660t^{15} + 5172389585290732t^{14} + 8032642364611869t^{13} + 8034989508286121t^{12} + 6363623526592659t^{11} + 4020679293727413t^{10} + 2035586999034796t^{9} + 669977607604833t^{8} + 59483933934797t^{7} - 193946025627t^{6} + 9168748852978t^{5} + 397940458800t^{4} - 5622728856t^{3} + 2108142106t^{2} + 11893509t + 83850 \right)D^{5} + \left( t + 1 \right) ^{2} \left( 12415493407530500t^{16} + 52525869607664040t^{15} + 98086961664377909t^{14} + 109002621146844517t^{13} + 83506332491110848t^{12} + 47000124673546106t^{11} + 20700430972782569t^{10} + 7482455683344473t^{9} + 1628972229026733t^{8} + 81472807155078t^{7} + 76097832031173t^{6} + 30670248761545t^{5} + 651078458236t^{4} + 19460023632t^{3} - 1994292587t^{2} - 12525351t - 41925 \right)D^{4} + t \left( t + 1 \right) \left( 52145072311628100t^{16} + 262877252528762620t^{15} + 585011593043711795t^{14} + 761760373665975649t^{13} + 652447509086893094t^{12} + 385832195117589758t^{11} + 160857178691993121t^{10} + 51358440874971159t^{9} + 13749264773158668t^{8} + 2317038856419428t^{7} + 442316777655033t^{6} + 314806991841263t^{5} + 53721369319030t^{4} + 917258402866t^{3} + 14891115155t^{2} + 122143129t + 37700 \right)D^{3} + 4 t ^{2} \left( 28803944705470760t^{16} + 169838346569044006t^{15} + 444630259052797777t^{14} + 682332915879850501t^{13} + 683828574421569954t^{12} + 467439321445155403t^{11} + 218073356011814252t^{10} + 69239854678325430t^{9} + 16844006285124245t^{8} + 4005815567812452t^{7} + 899737806853702t^{6} + 332651098468761t^{5} + 129091389167997t^{4} + 12583137993791t^{3} + 199109646705t^{2} + 4006451768t + 23450652 \right)D^{2} + 4 t ^{2} \left( 31287043386976860t^{16} + 181157110251350212t^{15} + 464375585899988775t^{14} + 694197364834413040t^{13} + 671801293889355467t^{12} + 435227115193564940t^{11} + 184389746808227140t^{10} + 48755153435480694t^{9} + 9542767579029866t^{8} + 2637412287739940t^{7} + 806213690768635t^{6} + 295952490611572t^{5} + 92940682776919t^{4} + 6177534140844t^{3} + 107607593650t^{2} + 2106424614t + 8988720 \right)D + 8 t ^{2} \left( 6385110895301400t^{16} + 36512746427221080t^{15} + 92233487295053982t^{14} + 135363172923922308t^{13} + 127806388466988012t^{12} + 79678028568714072t^{11} + 31371230083470947t^{10} + 7068977323134556t^{9} + 1133310054069566t^{8} + 413183375792662t^{7} + 150478513886594t^{6} + 50896025968550t^{5} + 13514392702418t^{4} + 642951416658t^{3} + 13258378913t^{2} + 224365570t + 670800 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ4P4 × Gr(2, 7)

|λ| = 16.1086 λ ≈ 16.1086; multiplicity 1; Jordan blocks 1 λ ≈ -7.871536 - 7.900707i; multiplicity 1; Jordan blocks 1 λ ≈ -7.871536 + 7.900707i; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 4; Jordan blocks 2, 2 λ ≈ 0.3172355 - 0.9402538i; multiplicity 1; Jordan blocks 1 λ ≈ 0.3172355 + 0.9402538i; multiplicity 1; Jordan blocks 1 4
Sector dimension
9
Spectral radius
$16.1086018$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$16.1086$11
$-7.871536 - 7.900707\mathrm{i}$11
$-7.871536 + 7.900707\mathrm{i}$11
$-1$42, 2
$0.3172355 - 0.9402538\mathrm{i}$11
$0.3172355 + 0.9402538\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{4} \left(\lambda^{5} - \lambda^{4} - 128 \lambda^{3} - 1922 \lambda^{2} + 1144 \lambda - 1973\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda^{5} - \lambda^{4} - 128 \lambda^{3} - 1922 \lambda^{2} + 1144 \lambda - 1973\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 8 & 84 & 42 & 48 & 80 & 128 & 340 \\ 1 & 0 & 4 & -8 & 0 & -8 & 60 & 96 & 32 \\ 1 & 0 & -1 & 24 & 16 & 8 & 48 & 60 & 48 \\ 0 & 1 & 2 & -1 & -1 & 0 & 0 & 8 & 24 \\ 0 & 1 & 0 & 4 & 1 & 4 & 0 & 0 & 24 \\ 0 & 0 & 1 & 0 & 1 & -1 & 16 & 24 & 12 \\ 0 & 0 & 0 & 1 & -\frac{1}{2} & 2 & -3 & -2 & 8 \\ 0 & 0 & 0 & 2 & \frac{7}{4} & 0 & 3 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 0\end{pmatrix}\notag\end{equation}$$