A note on affine toric varieties
A matrix \(D=(d_{ij})\) with non-negative integer entries defines a mapping of polynomial rings over a field \(k\) by \(x_i\to t_1^{d_{1i}}, \dots, t_m^{d_{mi}}\). The kernel of this mapping is the toric ideal associated with \(D\). The same matrix also defines a mapping \(Z^n\to Z^m\) whose kernel can be easily determined using the Smith normal form. The main result states that the toric set defined by \(D\) is identical to the variety defined by its toric ideal if and only if two conditions are satisfied, one of which can be formulated in terms of the elementary divisors of \(D\). This is then used (a) to obtain further results for algebraically closed \(k\), (b) to prove that monomial curves over arbitrary fields are affine toric varieties.
scientific article; zbMATH DE number 1547895