Let $k$ be a field of characteristic $0$ (not necessarily algebraically closed), and let $A=k[x^1,\ldots,x^m]/(f_1,\ldots,f_N)$ be an affine variety which is a complete intersection; i.e. $\dim(A)=m-N$. (For definiteness, let's assume that $m-N\geq 3$.)

Is it always possible to find an affine hypersurface $B=k[x^1,\ldots,x^m]/(c)$ such that the intersection with $A$ has dimention $m-N-1$ without adding any new singularities (i.e. compared to the singularities of $A$)?

  • 4
    $\begingroup$ this should follow from Bertini's theorem: the intersection with a general hyperplane does not add any new singularities. $\endgroup$
    – Hans
    Oct 14 '15 at 7:57
  • $\begingroup$ Thanks! Perhaps you could elaborate a bit more on it, and write it as an answer? What about fields that are not algebraically closed? $\endgroup$ Oct 14 '15 at 10:04
  • $\begingroup$ Does it matter that Bertini's theorem is formulated for projective varieties, when I'm interested in affine varieties? $\endgroup$ Oct 14 '15 at 10:28

This follows from Bertini's Theorem: see for example here. Note that your variety does not need to be projective. Let $\overline{k}$ be the algebraic closure of $k$. You know that there is an open dense subset of hyperplanes in $(\mathbb{P}_{\overline{k}}^n)^*$ such that the intersection with your variety does not add new singularities. Since the $k$-rational points of $(\mathbb{P}_{\overline{k}}^n)^*$ are dense, there is also a hyperplane defined over $k$.


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