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 $P \in \mathrm{dom}(f).$

  • $\begingroup$ The functions $f_i$ need to have the same degree. $\endgroup$
    – Youngsu
    Apr 7 '20 at 5:36
  • $\begingroup$ @Youngsu The functions f_i belong to k(X), so they are fractions of homogenous polynomials of the same degree by definition. $\endgroup$
    – user350031
    Apr 7 '20 at 6:42

I have also been reading chapters 1 and 2 of "The Arithmetic of Elliptic Curves" recently and from that point of view your definition is correct.

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.


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