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Given a non-constant rational map $\phi\colon E_1\to E_2$ between projective (irreducible) curves $E_1$ and $E_2$, we can define the pull-back $\phi^*\colon K(E_2)\to K(E_1)$. The degree of $\phi$ is the degree of the field extension $K(E_1)/\phi^* K(E_2)$. So far so good, but now my book is claiming that the degree of $\phi$ is given by the degree of the field extension $\overline K(E_1)/\phi^*\overline K(E_2)$ (where clearly they take $\phi^*\colon\overline K(E_2)\to\overline K(E_1)$). Are those degrees always the same? Otherwise I wouldn't know why my book is switching between them

Sha Vuklia
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  • Yes, the degree is invariant under base change, see e.g. https://math.stackexchange.com/questions/932354/degree-of-morphism-of-schemes. Does this answer your question? – Watson Jun 21 '21 at 19:04
  • @Watson I'm not familiar yet with schemes. Next year I'm taking a course on it, so I'll come back to this question then. Thanks for the link though! – Sha Vuklia Jun 21 '21 at 19:20

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