Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-80-23-c

·

Blow-up of a codimension-two linear section of $\operatorname{LG}(3,6)$ along a K3 surface of degree $16$.

  • $\pi_1\colon X\to Y$ is the blow-up of a smooth codimension-two linear section $Y$ of the Lagrangian Grassmannian $\mathrm{LG}(3,6)$ (1-256-57-a), a del Pezzo fourfold of index $2$ and degree $16$, along a K3 surface of genus $9$ and degree $16$, the intersection of $Y$ with two further hyperplanes.

  • $\pi_2\colon X\to\mathbb P^1$ is a fibration whose general fiber is a smooth prime Fano threefold of index one, degree $16$ and genus $9$.

This is a Fano fourfold of K3 type, appearing as case K3-26 in Bernardara–Fatighenti–Manivel–Tanturri.

Varieties in this family can be blown down to

Defining data
Ambient
ℙ1P1 × Gr(3, 6)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w3) + 2 copies of E^(w4) + E^(w1 + w4)
Bundle
wedge^2 U_2^v + 2 copies of O_2(1) + O_1(1) box O_2(1)
ZL62 label
153.332i
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;6) \\ \det\mathcal{E} &\cong \mathcal{O}(1;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 23
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 80
$\chi(X, \mathrm{T}_X)$ = -23

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 80
$\mathrm{c}_1^2\mathrm{c}_2$ = 104
$\mathrm{c}_2^2$ = 132
$\mathrm{c}_1\mathrm{c}_3$ = 20
$\mathrm{c}_4$ = 32
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 24 1 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 + 23 0 x0 + 23 x1
$\mathrm{HH}^{ 3 }$ = x1 + 16 0 0 x1 + 15 1
$\mathrm{HH}^{ 4 }$ = 24 0 0 1 0 23
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 18t + 42t^{2} + 18t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

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 $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 4t + 1 \right) \left( 5t + 1 \right) ^{3} \left( 16t^{2} + 8t - 1 \right) \left( 144t^{2} + 24t - 1 \right) \left( 96832512t^{6} + 69028608t^{5} + 16585472t^{4} + 1477808t^{3} + 16044t^{2} - 2032t + 5 \right)D^{6} + \left( 5t + 1 \right) ^{2} \left( 93702885212160t^{12} + 150220833030144t^{11} + 95349690335232t^{10} + 30600133902336t^{9} + 5117432586240t^{8} + 372955185152t^{7} - 3327504384t^{6} - 1794709888t^{5} - 70348816t^{4} - 3120580t^{3} - 78820t^{2} + 5991t - 10 \right)D^{5} + \left( 5t + 1 \right) \left( 3904286883840000t^{13} + 6333003458150400t^{12} + 4226538458382336t^{11} + 1502914475999232t^{10} + 305579141349376t^{9} + 34636402056192t^{8} + 1794656406016t^{7} - 5647079808t^{6} - 2938062016t^{5} + 134693756t^{4} + 15115708t^{3} + 107441t^{2} - 3596t + 5 \right)D^{4} + t \left( 81990024560640000t^{13} + 137724665167872000t^{12} + 98087247367372800t^{11} + 38632206979596288t^{10} + 9168955753611264t^{9} + 1327568275003392t^{8} + 110285106624000t^{7} + 4002364528256t^{6} - 60378452976t^{5} - 7792536780t^{4} - 45035004t^{3} + 3316773t^{2} - 86976t - 23 \right)D^{3} + 2 t ^{2} \left( 90579455705088000t^{12} + 142292067257548800t^{11} + 94718612313538560t^{10} + 34803488419872768t^{9} + 7678235731707904t^{8} + 1025897988470784t^{7} + 77261877151104t^{6} + 2339603152224t^{5} - 61223094768t^{4} - 5314314828t^{3} - 52423516t^{2} + 76803t - 25830 \right)D^{2} + 32 t ^{2} \left( 6149251842048000t^{12} + 9169337294438400t^{11} + 5784266209075200t^{10} + 2009162443333632t^{9} + 417152164885248t^{8} + 51976442226944t^{7} + 3563917923152t^{6} + 85575939380t^{5} - 3889046368t^{4} - 247635421t^{3} - 2444170t^{2} - 32400t - 510 \right)D + 64 t ^{2} \left( 1254949355520000t^{12} + 1803654107136000t^{11} + 1094154774036480t^{10} + 364431801212928t^{9} + 72212653711872t^{8} + 8505602790912t^{7} + 537083665376t^{6} + 9768211896t^{5} - 743210440t^{4} - 38795006t^{3} - 384336t^{2} - 6997t - 40 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × Gr(3, 6)

|λ| = 28.97056 λ ≈ 28.97056; multiplicity 1; Jordan blocks 1 λ ≈ 9.656854; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 3; Jordan blocks 1, 2 λ ≈ -4.970563; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 1; Jordan blocks 1 λ ≈ -1.656854; multiplicity 1; Jordan blocks 1 3
Sector dimension
8
Spectral radius
$28.9705627$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$28.97056$11
$9.656854$11
$-5$31, 2
$-4.970563$11
$-4$11
$-1.656854$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right) \left(\lambda + 5\right)^{3} \left(\lambda^{2} - 8 \lambda - 16\right) \left(\lambda^{2} - 24 \lambda - 144\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right) \left(\lambda + 5\right)^{2} \left(\lambda^{2} - 8 \lambda - 16\right) \left(\lambda^{2} - 24 \lambda - 144\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 56 & 144 & 384 & 1184 & 808 & 2240 \\ 1 & -1 & 4 & 48 & 40 & 312 & 126 & 512 \\ 1 & 1 & 4 & 16 & 56 & 144 & 114 & 296 \\ 0 & 1 & 1 & 3 & 4 & 48 & 16 & 78 \\ 0 & 0 & 2 & 2 & 4 & 16 & 16 & 36 \\ 0 & 0 & 0 & 2 & 1 & 3 & 2 & 12 \\ 0 & 0 & 0 & 0 & 4 & 4 & 0 & 8 \\ 0 & 0 & 0 & 0 & 0 & 4 & 2 & 0\end{pmatrix}\notag\end{equation}$$