On the simplification of singularities by blowing up at equimultiple centers

Resolution of singularities of varieties over fields of characteristic zero can be proved by using the multiplicity as main invariant. The proof of this result leads to new questions in positive characteristic. We discuss here results which follow by induction on the dimension of the varieties. Fix a variety $X^{(d)}$ of dimension $d$ over a {\em perfect field} $k$ or, more generally, a pure dimensional scheme of finite type over $k$. Fix a closed point $x\in X^{(d)}$ of multiplicity $e>1$. Define a local simplification of the multiplicity at $x\in X^{(d)}$ as a proper birational map, say $X^{(d)}\leftarrow X^{(d)}_1$, where $X^{(d)}$ denotes now a neighborhood of $x$, so that $X^{(d)}_1$ has multiplicity $<e$ at any point $x_1\in X^{(d)}_1$. Assume, by induction on $d$, the existence of local simplifications of the multiplicity for schemes over $k$ of dimension $d'$, for all $d' <d$. We prove, under this inductive assumption, that a local simplification at $x\in X^{(d)}$ can be constructed when $(C_{X,x})_{red}$ is not regular. Here $C_{X,x}$ denotes the tangent cone of $x\in X$, and $(C_{X,x})_{red}$ is the reduced scheme. The paper uses classical results of commutative algebra, and compares the effect of blowing up along equimultiple centers, and along normally flat centers.