Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-240-54-c

·

Blow-up of the degree-$5$ del Pezzo fourfold along a del Pezzo surface of degree $5$.

  • $\pi_1\colon X\to\mathrm{Gr}(4,7)$ is a closed embedding, with image of degree $25$.

  • $\pi_2\colon X\to W$ is the blow-up of the del Pezzo fourfold of degree $5$ (1-405-85-a) along a quintic del Pezzo surface, a codimension-two linear section of it.

  • $\psi\colon X\to\mathbb P^1$ is a del Pezzo fibration of degree $5$ whose fibers are the hyperplane sections of the degree-$5$ del Pezzo fourfold (1-405-85-a) through the quintic del Pezzo surface.

Varieties in this family can be blown up to

Varieties in this family can be blown down to

Defining data
Ambient
Fl(3, 4, 7)
node 1: fundamental weight ω_1node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
2 copies of E^(w1) + E^(w3) + E^(w4) + E^(w5)
Bundle
2 copies of U_3^v + O(1,0) + O(0,1) + Q_1^v ot U_2^v ot wedge^3 U_3^v
ZL62 label
60B.43200
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4,4) \\ \det\mathcal{E} &\cong \mathcal{O}(3,3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 54
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 240
$\chi(X, \mathrm{T}_X)$ = -4

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 240
$\mathrm{c}_1^2\mathrm{c}_2$ = 156
$\mathrm{c}_2^2$ = 101
$\mathrm{c}_1\mathrm{c}_3$ = 46
$\mathrm{c}_4$ = 13
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 7 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 + 4 0 x0 + 4 x1
$\mathrm{HH}^{ 3 }$ = x1 + 46 0 0 x1 - 10 56
$\mathrm{HH}^{ 4 }$ = 54 0 0 0 0 54
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 49t + 140t^{2} + 49t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(288)
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( 11t^{2} + 9t - 1 \right) \left( 389t^{6} + 81t^{5} - 60t^{4} + 156t^{3} - 23t^{2} - 11t - 1 \right) \left( 14821186560t^{13} + 84694384800t^{12} + 191811373536t^{11} + 216509277056t^{10} + 135704642340t^{9} + 56176006640t^{8} + 20000445535t^{7} + 3153564970t^{6} - 1437358146t^{5} - 228602952t^{4} + 23622284t^{3} + 1338712t^{2} + 15852t + 432 \right)D^{6} + \left( 1331817003095040t^{22} + 10157706643552320t^{21} + 32881004471847072t^{20} + 58767003960159840t^{19} + 64607151400188700t^{18} + 47884210359928332t^{17} + 27679266810314387t^{16} + 14369327977084139t^{15} + 6233062873201258t^{14} + 2015864209888485t^{13} + 609920904435690t^{12} + 118106955880409t^{11} - 55556120449411t^{10} - 24308759001655t^{9} + 2419518245685t^{8} + 926126617611t^{7} - 55540344562t^{6} - 538045394t^{5} + 768522248t^{4} - 127315240t^{3} - 5527612t^{2} - 44964t - 864 \right)D^{5} + \left( 11098475025792000t^{22} + 83636550282350400t^{21} + 266575850648083200t^{20} + 467369303587393184t^{19} + 502406538738797156t^{18} + 359775223803741316t^{17} + 192456912640311413t^{16} + 85207047423800371t^{15} + 28398457846310632t^{14} + 6000495178327588t^{13} + 1672567883239374t^{12} + 581375468607309t^{11} - 82864244255376t^{10} - 55215135354750t^{9} + 6763262640509t^{8} + 3050986787525t^{7} - 95667083996t^{6} - 39893510310t^{5} - 3716615236t^{4} + 91081124t^{3} + 5267732t^{2} + 63636t + 432 \right)D^{4} + t \left( 46613595108326400t^{21} + 347937194419939200t^{20} + 1093917428182554720t^{19} + 1884604055731977120t^{18} + 1989061384136761108t^{17} + 1398294175873865988t^{16} + 726029247687428693t^{15} + 294535718759843821t^{14} + 73912512441573894t^{13} + 1401030563198783t^{12} - 2085293704791540t^{11} + 305247851685955t^{10} - 240514485599912t^{9} - 132552757369777t^{8} - 1610193447325t^{7} + 4612170467545t^{6} - 59765153466t^{5} - 20666738758t^{4} - 1119950940t^{3} - 42700528t^{2} - 718196t - 108 \right)D^{3} + t ^{2} \left( 102993848239349760t^{20} + 763135551233075040t^{19} + 2372133882719751936t^{18} + 4025675628267611840t^{17} + 4184109756639291680t^{16} + 2906550748615536400t^{15} + 1493036856146122082t^{14} + 578683548147080154t^{13} + 113526827808735304t^{12} - 21520078670376065t^{11} - 13012676979302920t^{10} - 1124816653084680t^{9} - 388954174695472t^{8} - 184196517739311t^{7} - 9401129995162t^{6} + 5764377149264t^{5} + 256224691592t^{4} - 12523793512t^{3} - 981951888t^{2} - 45982404t - 566112 \right)D^{2} + 2 t ^{2} \left( 55936314129991680t^{20} + 412216856630852640t^{19} + 1270041315101141184t^{18} + 2129579512347169920t^{17} + 2186263454755156184t^{16} + 1505640555574779144t^{15} + 769970909646020596t^{14} + 290148852081314076t^{13} + 45931885929518396t^{12} - 18382238766224437t^{11} - 8763592223771542t^{10} - 872014164552508t^{9} - 146607577801184t^{8} - 71095363533792t^{7} - 3581545906520t^{6} + 2123970307132t^{5} + 169365653120t^{4} - 2481153680t^{3} - 275558528t^{2} - 13372320t - 118656 \right)D + 24 t ^{2} \left( 1902595718707200t^{20} + 13969399626189600t^{19} + 42773159655800640t^{18} + 71105797956688704t^{17} + 72347510243574248t^{16} + 49520003876387160t^{15} + 25265354571607144t^{14} + 9341873433284560t^{13} + 1233048786785792t^{12} - 746297507198261t^{11} - 317378761560338t^{10} - 30053746637082t^{9} - 3273811652280t^{8} - 1894257945142t^{7} - 64319610884t^{6} + 55225738900t^{5} + 5243762880t^{4} - 45130440t^{3} - 6083760t^{2} - 273960t - 1728 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(3, 4, 7)

|λ| = 10.09017 λ ≈ 10.09017; multiplicity 1; Jordan blocks 1 λ ≈ -6.905694 - 3.616011i; multiplicity 1; Jordan blocks 1 λ ≈ -6.905694 + 3.616011i; multiplicity 1; Jordan blocks 1 λ ≈ 2.721219; multiplicity 1; Jordan blocks 1 λ ≈ 0.5962467 - 1.333665i; multiplicity 1; Jordan blocks 1 λ ≈ 0.5962467 + 1.333665i; multiplicity 1; Jordan blocks 1 λ ≈ -1.102323; multiplicity 1; Jordan blocks 1 λ ≈ -1.09017; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$10.0901699$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$10.09017$11
$-6.905694 - 3.616011\mathrm{i}$11
$-6.905694 + 3.616011\mathrm{i}$11
$2.721219$11
$0.5962467 - 1.333665\mathrm{i}$11
$0.5962467 + 1.333665\mathrm{i}$11
$-1.102323$11
$-1.09017$11
$-1$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda^{2} - 9 \lambda - 11\right) \left(\lambda^{6} + 11 \lambda^{5} + 23 \lambda^{4} - 156 \lambda^{3} + 60 \lambda^{2} - 81 \lambda - 389\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda^{2} - 9 \lambda - 11\right) \left(\lambda^{6} + 11 \lambda^{5} + 23 \lambda^{4} - 156 \lambda^{3} + 60 \lambda^{2} - 81 \lambda - 389\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 6 & 6 & 6 & 15 & 3 & 4 & 4 & 10 \\ 1 & 0 & 2 & 2 & -4 & 2 & 3 & 12 & -4 \\ 1 & 0 & -1 & 4 & 10 & 2 & 0 & 0 & 4 \\ 0 & 1 & -1 & 0 & 2 & 0 & 0 & -6 & 3 \\ 0 & 1 & 3 & 0 & -1 & 0 & 2 & 8 & 0 \\ 0 & 2 & 2 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 5 & \frac{5}{3} & \frac{10}{3} & 0 & 4 & -2 \\ 0 & 0 & 0 & 1 & \frac{13}{3} & \frac{2}{3} & 0 & -1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 9 & 0\end{pmatrix}\notag\end{equation}$$