Toric varieties provide special examples of Zariski open affine varieties in \( \mathbb C^n \). Some examples are \( V(x^2 -y^3) \), or any such variety generated by binomials of the form \( V(x^{\alpha} – y^{\beta}) \).
A torus is defined by \( (\mathbb C^*)^n \), the cartesian product of non-zero complex numbers with the associated multiplicative structure. A torus acts on itself by multiplication. A toric variety is any affine variety \( V \) that contains such a torus and the toral action extends to \( V \).
This mix of group structure of a torus with an algebro-geometric structure of an affine variety make these useful objects for calculations, and a testing ground for theories in many different areas from complex geometry to topology. Associated to a torus \(T\) is its set of characters: group homomorphisms \(\chi^m: T \to \mathbb C^*\), all of which are of the form \(\chi^m(t) = t_1^{m_1}… t_n^{m_n}\) for \(m = (m_1,…, m_n)\), once \(T\) is identified with \((\mathbb C^*)^n\).
Clearly the set of characters are parametrized by a lattice in \(\)