I have just started reading about Toric varieties from David Cox and I have a couple of very basic questions about toric ideals and aff. toric varieties.
1) Eventually, a toric ideal can be defined as a prime ideal generated by differences of two monomials. I am a little confused with the definition of a monomial in this context. In all of the examples and all the proofs in the book, from what I can tell it is implied that the coefficient of the monomial should always be 1, meaning that $X$ is monomial but $2X$ isn't.
That seems a bit weird to me, as that would mean that $V(X-Y)$ is an affine toric variety and $V(X-2Y)$ isn't, unless we can somehow write $I=\langle X-2Y \rangle$ as an ideal generated by difference of monomials with coefficient one, which I don't see how. So is $V(X-2Y)$ not an affine toric variety?
2) Given that the ideal of an affine toric variety is a toric ideal, meaning that it is generated by binomials, I am confused as to what happens when you for example parallel translate an affine toric variety. For example, let $V=V(X-Y)$ an affine toric variety and translate it to $W=V+w$ where $w$ is any vector in $\mathbb R^2$, say $w=(0,2)$. Clearly $W$ is isomorphic to $V$ where the morphism is also a group homomorphism. Doesn't that however imply that $W$ is also an affine toric variety? But if that is the case, I fail to see how $I(W)=\langle X-Y+2\rangle$ is generated by binomials (i.e is a toric ideal).