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I want to prove that given a rational map $f:\mathbb{P}^n\dashrightarrow\mathbb{P^m}$ the locus where it is not defined $X\subset \mathbb{P}^n$ is a projective variety such that any irreducible component has dimension $\operatorname{dim}\ge n-m-1$ (we can assume $n\ge m+1$). Clearly if you can show that it is the zero locus of $m+1$ polynomials you conclude, and it seems indeed true that it is, see Rational Maps $\mathbb{P}^n \rightarrow \mathbb{P}^m$. Since my actual knowledge doesn't allow me to understand that proof I want to ask for some hints or solutions using only the theory covered by Hartshorne chapter 1.

Samuel Ascoli
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  • The hint at the link is referring to a proposition in Chapter II in Hartshorne: Proposition II.7.12. One may formulate and prove a similar proposition in the language of Chapter I in Hartshorne: If $Y$ is a quasi projective variety over an algebraically close field $k$, we may speak of invertible sheaves $L$ on $Y$ and the relation between maps $p: Y \rightarrow \mathbb{P}^m$ and global sections of $L$. Are you familiar with the notion "invertible sheaf" in the language of Chapter HH I? – hm2020 Feb 04 '25 at 11:54
  • I am not really familiar with sheaves of modules (they only appear in chapter 2) if this is what you refer to (on Hartshorne index they appear for the first time at page 109). – Samuel Ascoli Feb 04 '25 at 12:49

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