Continuity of intersection points of projective algebraic varieties
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Clash Royale CLAN TAG #URR8PPP up vote 1 down vote favorite Suppose that $X subset mathbbCP^n$ is a (possibly singular) algebraic curve. Let $H_tau subset mathbbCP^n$ be a family of hyperplanes, with $tau in (-1,1)$. By Bezout's theorem, the count of intersection points (with multiplicity) is independent of $tau$. (Let us assume that $H_tau$ does not contain any irreducible components of $X$, and that $X$ is reduced. This justifies the application of Bezout's theorem). Question: I would like to say that the intersection points vary continuously in $mathbbCP^n$, with respect to the Euclidean topology. Is this statement true, and if so, where can it be found in the literature? complex-analysis algebraic-geometry share | cite | improve this question edited Aug 6 at 20:06 asked Aug 6 at 18:51 user142700 1,190 5 15 1 what is the role of $tau$ here? You might need some kind of "generic" requirement. As in, take