# Definition of a rational map between projective varieties.

Let $X\subset\mathbb{P}^n$ and $Y\subset\mathbb{P}^m$ be projective varieties. If $m$ homogeneous polynomials in $n+1$ variables of the same degree give a partially defined map from $X$ to $Y$ then this map is called a rational map.

My question is why do some people insist in the definition of a rational map that $X$ be irreducible?

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## 1 Answer

Irreducible varieties have a function fields (the stalk at the generic point in scheme language), and it is a useful fact that rational maps correspond 1:1 to homomorphisms between the function fields. But of course the definition of a rational map doesn't require any assumption, and in fact, works for arbitrary varieties/schemes. A quite general treatment can be found in the Stacks Project, 24.10.

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