Separating sets, metric tangent cone and applications for complex algebraic germs

Abstract

Abstract. An explanation is given for the initially surprising ubiquity of separating sets in normal complex surface germs. It is shown that they are quite common in higher dimensions too. The relationship between separating sets and the geometry of the metric tangent cone of Bernig and Lytchak is described. Moreover, separating sets are used to show that the inner Lipschitz type need not be constant in a family of normal complex surface germs of constant topology.

Thin-thick decomposition for real definable isolated singularities

“…Indeed, this map is bi-Lipschitz on each pancake. The results of [1] and [5] imply thatΦ is bi-Lipschitz on each pancake. Since, a pancake decomposition is finite, the mapΦ is also finite.…”

“…or a choking horn [3], then there is a direction v ∈ L 0 X who is not a simple direction. Whenever the germ (X , 0) has a local separating set Y ⊂ X at 0 and v ∈ L 0 X , the direction v must belong to the singular locus of the tangent cone of Y at 0 (see [4,5]) and thus it cannot be a simple direction.…”

“…The first one, called the thick zone, contains "almost everything" and is "almost metrically conical". The other zone, called the thin zone, is not metrically conical, has density zero at the origin and moreover contains some of the most important invariants of the Lipschitz geometry of singular subset germs such as fast loops, choking-horns or separating sets ( [4,5,3]). The topology of the thin-thick decomposition is Lipschitz invariant and moreover the complete inner Lipschitz invariant of normal complex algebraic surface singularity germs can be obtained as the iterated thin-thick decomposition [7].…”

“…The topology of the components of the thick zone are related to the topology of the corresponding parts of the metric tangent cone (see [1] or [5]). Finally we show that the topology of the thin-thick decomposition of the set, equipped with the multiplicity, is an invariant for the blow-spherical equivalence.…”

Separating Sets on Semi-Weighted Homogeneous Hypersurface Singularities

“…they are not bi-Lipschitz homeomorphic to cones, even when such algebraic sets are equipped with the arc length metric (inner metric). Following some ideas presented there, it was possible to obtain other results about the bi-Lipschitz geometry of complex algebraic surface singularities (see [2][3][4]). One of the tools used to understand the bi-Lipschitz geometry of algebraic singularities was the notion of so-called separating sets (see [4]).…”