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Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map $f^{-1}:Y\to X$ is rational as well? Could you recommend any exact reference to a theorem which guarantees this?

edit Oct 30-31 I have looked carefully through the provided answers and references and still did not find any indication how to prove that the inverse map is rational. Theorem 12.83 in page 355 from [1] mentioned below does not contain the part of Proposition 2 below starting from 'hence an isomorphism onto its image'. Definition of an open immersion, see Definition 3.40 in page 83 from [1], does not say that an open immersion is an isomorphism onto its image. If there is a theorem that an open immersion in the sense of [1] is an isomorphism onto its image, then could you provide an exact reference to it (such a theorem seems to be more or less equivalent to the initial question)?

Related question: Isomorphism between varieties of char 0.

[1] U. Görz and T. Wedhorn, Algebraic Geometry I, Vieweg+Teubner Verlag-Springer Fachmedien Wiesbaden GmbH 2010.

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map $f^{-1}:Y\to X$ is rational as well? Could you recommend any exact reference to a theorem which guarantees this?

edit Oct 30-31 I have looked carefully through the provided answers and references and still did not find any indication how to prove that the inverse map is rational. Theorem 12.83 in page 355 from [1] mentioned below does not contain the part of Proposition 2 below starting from 'hence an isomorphism onto its image'. Definition of an open immersion, see Definition 3.40 in page 83 from [1], does not say that an open immersion is an isomorphism onto its image. If there is a theorem that an open immersion in the sense of [1] is an isomorphism onto its image, then could you provide an exact reference to it (such a theorem seems to be more or less equivalent to the initial question)?

Related question: Isomorphism between varieties of char 0.

[1] U. Görz and T. Wedhorn, Algebraic Geometry I, Vieweg+Teubner Verlag-Springer Fachmedien Wiesbaden GmbH 2010.

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map $f^{-1}:Y\to X$ is rational as well? Could you recommend any exact reference to a theorem which guarantees this?

edit Oct 30-31 I have looked carefully through the provided answers and references and still did not find any indication how to prove that the inverse map is rational. Theorem 12.83 in page 355 from [1] mentioned below does not contain the part of Proposition 2 below starting from 'hence an isomorphism onto its image'. Definition of an open immersion, see Definition 3.40 in page 83 from [1], does not say that an open immersion is an isomorphism onto its image. If there is a theorem that an open immersion in the sense of [1] is an isomorphism onto its image, then could you provide an exact reference to it such a theorem seems to be more or less equivalent to the initial question)?

Related question: Isomorphism between varieties of char 0.

[1] U. Görz and T. Wedhorn, Algebraic Geometry I, Vieweg+Teubner Verlag-Springer Fachmedien Wiesbaden GmbH 2010.

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map $f^{-1}:Y\to X$ is rational as well? Could you recommend any exact reference to a theorem which guarantees this?

edit Oct 30-31 I have looked carefully through the provided answers and references and still did not find any indication how to prove that the inverse map is rational. Theorem 12.83 in page 355 from [1] mentioned below does not contain the part of Proposition 2 below starting from 'hence an isomorphism onto its image'. Definition of an open immersion, see Definition 3.40 in page 83 from [1], does not say that an open immersion is an isomorphism onto its image. If there is a theorem that an open immersion in the sense of [1] is an isomorphism onto its image, then could you provide an exact reference to it (such a theorem seems to be more or less equivalent to the initial question)?

Related question: Isomorphism between varieties of char 0.

[1] U. Görz and T. Wedhorn, Algebraic Geometry I, Vieweg+Teubner Verlag-Springer Fachmedien Wiesbaden GmbH 2010.

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map $f^{-1}:Y\to X$ is rational as well? Could you recommend any exact reference to a theorem which guarantees this?

edit Oct 30-31 I have looked carefully through the provided answers and references and still did not find any indication how to prove that the inverse map is rational. Theorem 12.83 in page 355 from [1] mentioned below does not contain the part of Proposition 2 below starting from 'hence an isomorphism onto its image'. Definition of an open immersion, see Definition 3.40 in page 83 from [1], does not say that an open immersion is an isomorphism onto its image. If there is a theorem that an open immersion in the sense of [1] is an isomorphism onto its image, then could you provide an exact reference to it (the theorem seems to be more or less equivalent to the initial question)?

Related question: Isomorphism between varieties of char 0.

[1] U. Görz and T. Wedhorn, Algebraic Geometry I, Vieweg+Teubner Verlag-Springer Fachmedien Wiesbaden GmbH 2010.

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map $f^{-1}:Y\to X$ is rational as well? Could you recommend any exact reference to a theorem which guarantees this?

edit Oct 30-31 I have looked carefully through the provided answers and references and still did not find any indication how to prove that the inverse map is rational. Theorem 12.83 in page 355 from [1] mentioned below does not contain the part of Proposition 2 below starting from 'hence an isomorphism onto its image'. Definition of an open immersion, see Definition 3.40 in page 83 from [1], does not say that an open immersion is an isomorphism onto its image. If there is a theorem that an open immersion in the sense of [1] is an isomorphism onto its image, then could you provide an exact reference to it such a theorem seems to be more or less equivalent to the initial question)?

Related question: Isomorphism between varieties of char 0.

[1] U. Görz and T. Wedhorn, Algebraic Geometry I, Vieweg+Teubner Verlag-Springer Fachmedien Wiesbaden GmbH 2010.