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.
2
-
1$\begingroup$ What is the 'original image of $\xi$ in $X\times_k X$' ? $\endgroup$– Damian RösslerCommented Feb 25, 2014 at 10:33
-
$\begingroup$ 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. $\endgroup$– varCommented Feb 25, 2014 at 11:45
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
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.