Suppose $X$ is a projective scheme (or you can only consider the hypersurface case) in $\mathbb{P}^n$ over an algebraic closed field $k$, and $K/k$ is a field extension. Let $\xi$ be a point (neither closed point nor generic point) in $X$. Does the multiplicity of $x$ change under the base change from $k$ to $K$? I mean the multiplicities of the original images of $\xi$ in $X\times_kK$. Here we use the multiplicity defined by Hilbert-Shamuel funtion.
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1$\begingroup$ What is the 'original image of $\xi$ in $X\times_k X$' ? $\endgroup$– Damian RösslerFeb 25, 2014 at 10:33
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$\begingroup$ Suppose $p:X\times_kK\rightarrow X$ is the canonical projection, I mean $p^{-1}(\xi)$. Sorry I didn't explain it clearly in my original problem. $\endgroup$– varFeb 25, 2014 at 11:45
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Take a irreducible component of $\xi\times_kK$. If $k$ is perfect, the multiplicity of any of the irreducible component of $\xi\times_kK$ is not change.