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 $A$)? 



Singularities of complete intersections of affine varieties with hypersurfaces
This follows from Bertini's Theorem: [see for example here](https://www.encyclopediaofmath.org/index.php/Bertini_theorems).
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 $k$.


