Let C be a germ of curve singularity embedded in (kn, 0).
It is well known that the blowing-up of C centred on its closed ring, Bl(C),
is a finite union of curve singularities. If C is reduced we can iterate this process and, after a finite number
of steps, we find only non-singular curves. This is the desingularization process.
The main idea of this paper is to linearize the blowing-up of curve singularities
Bl(C) → C. We perform this by studying the structure of
[Oscr ]Bl(C)/[Oscr ]C as W-module,
where W is a discrete valuation ring contained in [Oscr ]C.
Since [Oscr ]Bl(C)/[Oscr ]C is a torsion
W-module, its structure is determined by the invariant factors of [Oscr ]C
in [Oscr ]Bl(C). The
set of invariant factors is called in this paper as the set of micro-invariants of C (see
Definition 1·2).In the first section we relate the micro-invariants of C to the Hilbert function of
C (Proposition 1·3), and we show how to compute them from the Hilbert function
of some quotient of [Oscr ]C (see Proposition 1·4).The main result of this paper is Theorem 3·3 where we give upper bounds of the
micro-invariants in terms of the regularity, multiplicity and embedding dimension.
As a corollary we improve and we recover some results of [6]. These bounds can be
established as a consequence of the study of the Hilbert function of a filtration of
ideals g = {g[r,i+1]}i [ges ] 0
of the tangent cone of [Oscr ]C (see Section 2). The main property of
g is that the ideals g[r,i+1] have initial degree
bigger than the Castelnuovo–Mumford regularity of the tangent cone of [Oscr ]C.Section 4 is devoted to computation the micro-invariants of branches; we show how
to compute them from the semigroup of values of C and Bl(C) (Proposition 4·3).
The case of monomial curve singularities is especially studied; we end Section 4 with
some explicit computations.In the last section we study some geometric properties of C that can be deduced
from special values of the micro-invariants, and we specially study the relationship of
the micro-invariants with the Hilbert function of [Oscr ]Bl(C). We end the paper studying
the natural equisingularity criteria that can be defined from the micro-invariants
and its relationship with some of the known equisingularity criteria.
Complete Intersection Monomial Curves and the Cohen—Macaulayness of Their Tangent Cones
