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Let $X$ be smooth surface, such that $f:X\dashrightarrow \Bbb{A}^1\subset \Bbb{P}^1$ is a rational map, we have the standard result that the rational map into projective space has undetermined locus of codimension $\ge 2$.

However by definition over the undetermined point $p\in X$ there exist a $f_1,f_2 \in \mathcal{O}_{X,p}$ such that $f_2(p) = 0$ and $f_1/f_2 = f$ locally, since the vanishing locus of $f_2$ say $V(f_2)$ has codimension 1, does this means that undetermined locus is codimension 1.

I know there must be something wrong since the undetermined locus is codimension $\ge 2$, not 1.

yi li
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  • Sorry I ask a siily question example $x_1/x_2$ over $\Bbb{A}^1$ has undetermined locus codimension 1, but for projective space, we add some "infinity", so there are more defined points, it reduced the codimension to $\ge 2$ – yi li May 09 '23 at 13:13
  • that just like what we did in Riemann surface that meromorphic function is indeed holomorphic into Riemann sphere as the dimension drop 2 to (-1) – yi li May 09 '23 at 13:22
  • There's a related question from a number of years ago that might help: https://math.stackexchange.com/questions/1215990/any-rational-map-can-be-extended-to-codimension-one – KReiser May 09 '23 at 13:33

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