# What is the Newton polyhedron of a toric variety?

First of all I want to say that my knowledge of toric geometry is minimal.

A paper I'm reading considers a toric variety $X$ and then claims that in its Newton polyhedron, $\Delta(X)$, the fixed points of the torus action on $X$ correspond to the vertices of $\Delta(X)$ and the torus invariant lines (*) to the edges of $\Delta(X)$.

Somehow I've come to think of $\Delta(X)$ as the fan of $X$, but this is a recent paper and I doubt they'd use some nonstandard (?) name for the fan of $X$. Hence the question. Needless to say my google searches have been fruitless.

Regarding (*): I don't think that a priori $X$ has to contain any "lines" (either affine or projective), so I'm taking the phrase "torus invariant line" to mean the torus invariant curves joining to vertices and the term "line" coming from the fact that it corresponds to a line in $\Delta(X)$. Is this correct?

The last interpretation is correct, but I think these curves are isomorphic to $\Bbb P^1$ so indeed it would be correct to call these curves "lines", in the sense they are isomorphic to projective lines.

There is another construction of toric varieties which starts from a lattice polytope $P$ instead of a fan $F$. The usual orbit-cone correspondance for fan is inverted. For example, we have a bijective correspondance between vertices of $P$, full-dimensional cones in the fan $F$ and fixed points of the action. In particular, the part you read about invariant curves is just the usual orbit-cone correspondance.

What is this construction ? By definition point in $m \in P \cap \Bbb Z^k$ will be monomial $x^m$ and consider the map $f : T = (\Bbb C^*)^k \to \Bbb P^N, x \mapsto (x^{m_0}, \dots, x^{m_N})$. We define $X_P = \overline{f(T)}$.

This makes sense because in the usual construction the fan $F$ lives in $N$ but the monomials $m$ live in $M = N^{\vee}$. So we can't really say that fan and Newton polyhedron are the same since they live in a different lattice. On the other hand, if you take the normal fan of $P \subset M$ you will indeed land in $N$ and this is the correct construction.

More information can be found in the book "Toric Varieties" by Cox, Schenck and Little.

Edit : here is an example. We take $P$ to be the triangle with vertices $(0,0), (0,d)$ and $(d,0)$ for some $d >0$. This map is exactly the Veronese embedding, so we have $X_P \cong \Bbb P^2$. Taking the normal fan of the triangle give the usual fan for $\Bbb P^2$.

( There are interesting math behind it, as the polygon is the data of a line bundle corresponding to the map, and multiplying the polygon by a constant is exactly twisting this line bundle.)

• Thanks for your answer. As final question, do yu have any idea why would they use the term Newton polyhedron? Even in the book you mention the term is barely used, perhaps this is the name it goes by in the original russian sources? Jul 2, 2017 at 22:08
• @user347489 One can associate a Newton polygon to polynomials in two variables. Newton actually used these in studying their number of zeros. I don't know more of the history, though. Jul 2, 2017 at 22:37
• @user347489 : this is exactly it, the Newton polyhedron of $f(x) = \sum_i a_ix^i$ is the convex hull of the $i \in \Bbb Z^n$ with $a_i \neq 0$. There is a very nice theorem giving you number of solutions of the system of equation $f_1 = \dots = f_n = 0$ in term of volume of Newtwon polyhedron.
– user171326
Jul 3, 2017 at 7:28