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Let $f:U \to V$ be a flat, quasi-finite, surjective morphism between two affine varieties defined over $\mathbb{C}$. Assume that every closed fiber is reduced. Consider the function $\eta$ that sends a closed point $v \in V$ to the cardinality of the fiber $f^{-1}(v)$ over $v$. Is this function semi-continuous? If so, is it upper or lower semi-continuous? Recall, if we add the assumption that $f$ is proper, then it is well-known that $\eta$ is locally-constant.

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1 Answer 1

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The function is lower semi-continuous, see EGA4, Prop. 15.5.1.

Note that since you assume the fibres reduced hence (because the characteristic is 0) geometrically reduced, the map $f$ is in fact étale; and closed points have residue field $\mathbb{C}$ which is algebraically closed; hence your $\eta(v)$ is indeed the function $n(v)$ from EGA4.

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