# Confusion about definitions of rational map between projective varieties

I've been learning some algebraic geometry from a combination of:

1. Chapters 1 and 2 of Silverman's Arithmetic of Elliptic Curves,

2. Reid's Undergraduate AG

3. Hulek's Elementary AG

and I'm a bit confused about the definitions of "rational map between projective varieties".

QUESTION. I would like to know whether the following is a correct definition.

If so, it looks like rational maps $$X\dashrightarrow Y$$ are elements of $$\mathbb{P}^{n}_{k(X)}$$ - is this just a coincidence?

Let $$X\subseteq \mathbb{P}^m$$ and $$Y\subseteq \mathbb{P}^n$$ be irreducible closed subspaces.

We define an equivalence relation on tuples $$(f_0,\ldots,f_n)$$ where

1. each $$f_i \in k(X),$$
2. not all the $$f_i$$ are zero,

as follows:

$$(f_0,\ldots,f_n)\sim (g_0, \ldots, g_n) \; \; \; \Leftrightarrow \; \; \text{ there exists } h \in k(X) \text{ such that } f_i=g_ih \text{ for all } i.$$ The equivalence class of $$(f_0,\ldots,f_n)$$ will be denoted by $$[f_0:\ldots:f_n].$$

We say $$f=[f_0:\ldots:f_n]$$ is regular at $$P \in X$$ is it has a representative $$(g_0,\ldots,g_n)$$ such that

1. each $$g_i$$ is regular at $$P,$$

2. some $$g_i(P)$$ is non-zero.

In this case, we define $$f(P)$$ to be $$(g_0(P):\ldots:g_n(P)) \in \mathbb{P}^n.$$

The domain of $$f$$ is defined to be the set $$\mathrm{dom}(f)=\{P \in X \, : \, f \text{ is regular at } P\}.$$

We say $$f$$ is a rational map $$X\dashrightarrow Y$$ if $$f(P) \in Y$$ for all $$P \in \mathrm{dom}(f).$$

• The functions $f_i$ need to have the same degree. Apr 7 '20 at 5:36
• @Youngsu The functions f_i belong to k(X), so they are fractions of homogenous polynomials of the same degree by definition. Apr 7 '20 at 6:42

I would not call it a coincidence that it looks like rational maps are elements of $$\mathbb{P}_{K(X)}^n$$, however i think the author just wanted to make all the details clear to readers who are new to the field.