Tag 0283
\(U^{3} \oplus \langle -6 \rangle\)
U^3 + <-6>·Beauville-Bogomolov lattice of Kum_2
Geometric objects
Invariants
| Rank | 7 |
|---|---|
| Signature \((n_+, n_-)\) | (3, 4) |
| Determinant | \(6\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{7}\).
In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}, e_{6}\}\) \(\oplus\) \(\mathbb{Z}\{e_{7}\}\), an orthogonal sum.
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | |
|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| \(e_{2}\) | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{3}\) | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| \(e_{4}\) | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| \(e_{5}\) | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| \(e_{6}\) | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| \(e_{7}\) | 0 | 0 | 0 | 0 | 0 | 0 | -6 |
[[0, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, -6]]
Integral lattice
| Parity | even |
|---|---|
| Discriminant group | \(\mathbb{Z}/6\) |
| Unimodular | no |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 1 |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2, 3\}\) |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 6)\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 6)\, L^\Sigma(s - 3, \chi_{-6n})\) |
| Genus symbol | \(\mathrm{II}_{3,4}\ \bigl(2\colon 1^{-6}\,[2^{1}]_{1};\ 3\colon 1^{-6}\,3^{1}\bigr)\) |
| Class number of the genus | 1 |
| Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\) | 3 |
Orbits of primitive vectors \(F_{L,\Gamma}(z, w) = \sum_n c_\Gamma(n)\, z^n\), with \(w^n\) for \(z^{-n}\)
| \(\Gamma = \widetilde{O}(L)\) | \(1 + 2\,z^{2} + w^{2} + z^{4} + w^{4} + O(z^{5}, w^{5})\) |
|---|---|
| \(\Gamma = \widetilde{SO}(L)\) | \(1 + 2\,z^{2} + w^{2} + z^{4} + w^{4} + O(z^{5}, w^{5})\) |
| \(\Gamma = \widetilde{O}^+(L)\) | \(1 + 2\,z^{2} + w^{2} + z^{4} + w^{4} + O(z^{5}, w^{5})\) |
| \(\Gamma = \widetilde{SO}^+(L)\) | \(1 + 2\,z^{2} + w^{2} + z^{4} + w^{4} + O(z^{5}, w^{5})\) |
| \(\Gamma = \mathrm{O}(L)\) | \(1 + 2\,z^{2} + w^{2} + z^{4} + w^{4} + O(z^{5}, w^{5})\) |
| \(\Gamma = \mathrm{SO}(L)\) | \(1 + 2\,z^{2} + w^{2} + z^{4} + w^{4} + O(z^{5}, w^{5})\) |
| \(\Gamma = \mathrm{O}^+(L)\) | \(1 + 2\,z^{2} + w^{2} + z^{4} + w^{4} + O(z^{5}, w^{5})\) |
| \(\Gamma = \mathrm{SO}^+(L)\) | \(1 + 2\,z^{2} + w^{2} + z^{4} + w^{4} + O(z^{5}, w^{5})\) |
Indefinite lattice
| Isotropic | yes |
|---|
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{-6, 2\}\) |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(L\) |
| Rank of \(R(L)\) | 7 |
| \([L : R(L)]\) | 1 |
| \(R(L)\) is primitive in \(L\) | yes |
| Proof | Each \(r_i\) 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 \(L\).
| Root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(r_{1}\) | \(e_{7}\) | \(-6\) |
| \(r_{2}\) | \(e_{1} + e_{2}\) | \(2\) |
| \(r_{3}\) | \(e_{3} + e_{4}\) | \(2\) |
| \(r_{4}\) | \(e_{5} + e_{6}\) | \(2\) |
| \(r_{5}\) | \(e_{1} + e_{2} + e_{3}\) | \(2\) |
| \(r_{6}\) | \(e_{1} + e_{2} + e_{5}\) | \(2\) |
| \(r_{7}\) | \(e_{1} + e_{3} + e_{4}\) | \(2\) |
Notes
\(H^2(Y, \mathbb{Z})\) with the Beauville-Bogomolov form, for \(Y\) deformation equivalent to the generalized Kummer variety \(K_{2}(T)\) of a complex torus \(T\) of dimension 2: \(U^3 \oplus \langle -6 \rangle\) (Rapagnetta).
Morphisms
References
- 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.