Isosingular Loci and the Cartesian Product Structure of Complex Analytic Singularities

Abstract

Abstract. Let X be a (not necessarily reduced) complex analytic space, and let F be a germ of an analytic space. The locus of points q in X at which the germ Xq is complex analytically isomorphic to F is studied. If it is nonempty it is shown to be a locally closed submanifold of X, and X is locally a Cartesian product along this submanifold. This is used to define what amounts to a coarse partial ordering of singularities. This partial ordering is used to show that there is an essentially unique way to completely decompose an arbitrary reduced singularity as a cartesian product of lower dimensional singularities. This generalizes a result previously known only for irreducible singularities.

Untitled

“…All the other charts look quite similar (you may compute the equations yourself). It can be seen that the strict transform S (8) is smooth and in each chart the exceptional curves are smooth and meet pairwise transversally. Hence we are finished (except for the normal crossings of the strict transform and the exceptional divisor, which can be obtained by a line blowup).…”

“…Similarly D 8 = {z = 0} and D 2 = {y = 0} intersect the strict transform S (8) zyzz(y+1)z given by V (x 2 + y) transversally. On the strict transform S (8) zyzz ( …”

“…Fix a point p ∈ X. To study the local nature of X at p one considers arbitrarily small open (Euclidean) neighbourhoods of p. This corresponds to taking the germ of X at p. A theorem of Ephraim [8] states that, in any dimension, the locus Y of points q ∈ X such that the germs of X in these points q are analytically isomorphic to the germ of X in p is a submanifold of X. For surfaces, Y can be the entire variety X (if it is smooth), or a collection of curves and/or points.…”

“…This implies that our resolution is a minimal resolution. 8 In the first part of this section we already remarked in which charts which exceptional divisors intersect. Now the results will be collected: the first charts in which exceptional curves intersect are (after the 4th blowup) the zyyy-chart and the zyyz-chart.…”

“…Explicitly, the four relevant total transforms are: in the zyzy(z +1)y-chart the exceptional curves D 7 = {y = 0} and D 3 = {z = 0} intersect transversally on the strict transform S (8) zyzy(z+1)y given as V (x 2 + z). Similarly D 8 = {z = 0} and D 2 = {y = 0} intersect the strict transform S (8) zyzz(y+1)z given by V (x 2 + y) transversally. On the strict transform S (8) zyzz ( …”