Lemma 53.2.2 (0BXZ)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 53: Algebraic Curves
Section 53.2: Curves and function fields
Lemma 53.2.2 (cite)

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Lemma 53.2.2. Let $k$ be a field. Let $X$ be a normal curve and $Y$ a proper variety. The set of rational maps from $X$ to $Y$ is the same as the set of morphisms $X \to Y$ .

Proof. A rational map from $X$ to $Y$ can be extended to a morphism $X \to Y$ by Lemma 53.2.1 as every local ring is a discrete valuation ring (for example by Varieties, Lemma 33.43.8). Conversely, if two morphisms $f,g: X \to Y$ are equivalent as rational maps, then $f = g$ by Morphisms, Lemma 29.7.10. $\square$

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