On the simplification of singularities by blowing up at equimultiple centers

TL;DR

This paper investigates the process of simplifying singularities of algebraic varieties by blowing up at equimultiple centers, using induction on dimension and analyzing tangent cones, with implications for characteristic zero and positive characteristic cases.

Contribution

It establishes a method for local simplification of singularities via blowing up at equimultiple centers under an inductive framework based on tangent cone regularity.

Findings

Local simplification is possible when the reduced tangent cone is not regular.

The approach compares blowing up along equimultiple and normally flat centers.

Results rely on classical commutative algebra and induction on dimension.

Abstract

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.