I have recently got interested in toric varieties and I have a question concerning their ideals.
Let $A \in \mathbb{Z}^{m \times n}$ and $\ker A = \{ u \in \mathbb{Z}^n \; | \; Au = 0 \}$. For any $u \in \ker A$ vectors $u_+$ and $u_-$ can be defined as follows: $$u_+ = \sum_{u_i > 0}u_ie_i \textrm{ and } u_- = -\sum_{u_i<0}u_ie_i.$$ The ideal $I_A = \langle \{ x^{u_+} - x^{u_-} \; | \; u \in \ker A \} \rangle$ is no doubt toric. The well-known way to construct $I_A$ is the following one. Let $L = \{ l^1,...,l^r \}$ be a basis for $\ker A$ and $$ I_L = \langle \{ x^{l^i_+} - x^{l^i_-} \; | \; i = \overline{1,r} \} \rangle $$ then $$ I_A = I_L : (x_1 \cdots x_n)^\infty. $$ Is it right that $I_L$ is always $I_A$-primary (except for the cases when $I_L$ is principal or $L \in \mathbb{N}^{n \times r}$ - then $I_L = I_A$), i.e. $I_A = \sqrt{I_L}$? If so is there any relatively easy proof of it? If not what is the case when this statement may fail?
In addition to that:
Are there any reasonable bounds on the size of a reduced Groebner basis of $I_A$ (for example for grevlex ordering)?
Any hint or reference would be appreciated.
Thanks in advance.