Tag 0287

\(U^{3} \oplus \langle -2 \rangle^{2}\)

U^3 + <-2>^2·Beauville-Bogomolov lattice of OG6

Invariants

Rank8
Signature \((n_+, n_-)\)(3, 5)
Determinant\(-4\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{8}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}, e_{6}\}\) \(\oplus\) \(\mathbb{Z}\{e_{7}\}\) \(\oplus\) \(\mathbb{Z}\{e_{8}\}\), an orthogonal sum.

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)
\(e_{1}\)01000000
\(e_{2}\)10000000
\(e_{3}\)00010000
\(e_{4}\)00100000
\(e_{5}\)00000100
\(e_{6}\)00001000
\(e_{7}\)000000-20
\(e_{8}\)0000000-2
 
[[0, 1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, 0, -2]]

Integral lattice

Parityeven
Discriminant group\((\mathbb{Z}/2)^{2}\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)2
Nikulin invariants \((r, a, \delta)\)\((8, 2, 1)\)
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{4} \det L\)\(\chi_{-4} = \left(\frac{-4}{\cdot}\right)\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 7)\, L^\Sigma(s - 4, \chi_{-4})}{L^\Sigma(s - 3, \chi_{-4})}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 7)}{L^\Sigma(s - 3, \chi_{-4})}\)
Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\)\(\geq 3\), by an embedding of \(U^{3}\) in the morphism files

Indefinite lattice

Isotropicyes

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{-2, 2\}\)
\(R(L) = \mathbb{Z}\Phi(L)\)\(L\)
Rank of \(R(L)\)8
\([L : R(L)]\)1
\(R(L)\) is primitive in \(L\)yes
ProofEach \(r_i\) in roots that generate \(\mathbb{Z}\Phi(L)\) is a root, and the \(r_i\) generate \(L\).

Roots that generate \(\mathbb{Z}\Phi(L)\)

The \(r_i\) are a basis of \(L\).

RootCoordinates\(b(r, r)\)
\(r_{1}\)\(e_{7}\)\(-2\)
\(r_{2}\)\(e_{8}\)\(-2\)
\(r_{3}\)\(e_{1} + e_{2}\)\(2\)
\(r_{4}\)\(e_{1} + e_{7}\)\(-2\)
\(r_{5}\)\(e_{3} + e_{4}\)\(2\)
\(r_{6}\)\(e_{3} + e_{7}\)\(-2\)
\(r_{7}\)\(e_{5} + e_{6}\)\(2\)
\(r_{8}\)\(e_{5} + e_{7}\)\(-2\)

Notes

\(H^2(Y, \mathbb{Z})\) with the Beauville-Bogomolov form, for \(Y\) deformation equivalent to O’Grady’s variety of dimension 6: \(U^3 \oplus \langle -2 \rangle^2\) (Rapagnetta).

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice. Row 9 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G = U(2)^3 \oplus \langle -4 \rangle^2\), which is this lattice twisted by 2, for the group \(\mathfrak{G}_9 \cong C_2^3\) of Table 10.2.

  • 02BN \(\Lambda_{\mathfrak{G}_{9}}(-1)\) A lattice \(M\) with \(M(-1) \cong L(2)^\perp\) in \(\mathrm{II}_{3,19}\), where \(L(2) = \Lambda^G\) for the group \(\mathfrak{G}_{9}\) of Table 10.2 of Hashimoto.
  • 027E \(U^{3} \oplus E_{8}(-1)^{2}\) The K3 lattice \(\mathrm{II}_{3,19}\), in which \(L(2)\) is the invariant lattice \(\Lambda^G\) of the group \(\mathfrak{G}_{9}\) of Table 10.2 of Hashimoto.

Morphisms

References

  1. A. Rapagnetta, "On the Beauville form of the known irreducible symplectic varieties", Math. Ann. 340 (2008), 77-95, arXiv:math/0606409, the table of the introduction.
  2. K. Hashimoto, "Finite symplectic actions on the K3 lattice", arXiv:1012.2682, Tables 10.2 and 10.3.