on projective toric varieties

Question

I would like to understand better the embedding of projective toric varieties on projective spaces.

If I start with $X$ a projective topic variety, for being projective we know that there exists an ample line bundle $L$ on $X$ whose sections provide a map $f: X \to \mathbb{P}^n$. I have two questions:

1. Being $X$ toric, can this $f$ be monomial?
2. Being monomials, since they are sections of the same twist of $L$ can I assume that they are of the same degree?

Thank you for any help.

Answer 1

A normal toric variety $X$ is a projective toric variety (i.e. has a closed torus invariant embedding $X\to \mathbb P^n$ for some $n$) if and only if its fan $\Sigma_X$ is the normal fan of some (I believe full-dimensional) polytope $\Delta$ in the $M_X$-lattice of the characters of $X$. In that case you may find a unique toric invariant Cartier divisor $D$ in $X$ such that the the global sections of $\mathcal O_X(D)$ are spanned by the characters in $\Delta$. The sheaf $\mathcal O_X(D)$ is ample, and it will give you your embedding into projective space if you twist it enough. The minimal amount you have to twist it bounded by some formula which involves the dimension of $X$, and in dimension $2$ you does not have to twist it at all. In any case, the global sections of $\mathcal O_X(D)^{\otimes k}$ will be spanned by the characters in the scaled polytope $k\Delta$ and the normal fan of $k\Delta$ will still be $\Sigma_X$. So let us assume that $\Delta$ is already huge enough so that $D$ is a very ample divisor. The embedding into projective space is now defined as follows. You list all the characters $m_0,...,m_r$ in $\Delta\cap M_X$ and define the morphism $f:X\to \mathbb P^r$ via $f(x)=(\chi^{m_0}(x):\chi^{m_1}(x):...:\chi^{m_r}(x))$.

So to answer your first question: the $f$ will be indeed monomial. To answer your second question: the monomials will in general not be of the same degree, since it is pretty easy to draw full-dimensional polytopes in $M$-space which contain monomials of different degree. The toric variety of the normal fan of such a polytope will then have an embedding into projective space whose entries are the monomials of that polytope (or some scaled version of it).

I hope this helps and I did not make a mistake. Everything I said should be also in the big book by Cox and co.