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Let $Y$ be a scheme of finite type over an algebraically closed field $k$. Show that the function $\phi(y) = dim_k(m_y/{m_y}^2)$ is upper semicontinuous on the set of closed point of $Y$ (i.e. for any point $y$, there exists an open neighborhood $U$, such that for any $x \in U, \phi(y) \geq \phi(x)$ ).

I have two thoughts about this problem:

1) If $y$ is a smooth point, then using the property that singular set is closed, one can show semicontinuity. So the difficulty comes from the singular point.

2) I would like to using semicontinuity theorem of cohomology of fibers, but I don't know how to construct the corresponding coherent sheaf.

2 Answers 2

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Since the question is local you may assume that $Y=Spec(A)\subset \mathbb A^N_k $, where $A=k[X_1,...,X_N]/(f_1,...,f_m)$.
In other words $Y$ is the fiber of $0$ for the morphism $f=(f_1,...,f_m):\mathbb A^N_k \to \mathbb A^m_k$ .
As in good old calculus we have a Jacobian matrix with value for each closed $x\in Y$ : $J(x)= (Jac (f))(x) =(\frac{\partial f_i }{\partial x_j}(x)) \quad (i=1,...,m \;; j=1,...,N) $
The number you are interested in is exactly the nullity of that matrix:
$ \phi(x)=dim(T_x(Y))=dim_k( ker\:J(x))=m-rank (J(x)) $
The conclusion follows : if $\phi(y)=d$, then some $(m-d)\times (m-d)$ minor of $J(y)$ is $\neq 0$.
It will remain $\neq0$ for all $x$ in a neighbourhood of $y$ and thus in that neighbourhood we will
have $rank (J(x))\geq m-d$ so that $\phi(x)=m-rank (J(x))\leq d$ as desired.

Note that neither (non)-singularity nor cohomology are evoked.

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    I thank QiL for his usual clarity. My question was really stupid.2012-04-24
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I think you can apply semicontinuity to sheaf of differentials.

But I don't see why affine line over C is a counterexample to andreas's question.