On Delta for parameterized Curve Singularities

Abstract

We consider families of parameterizations of reduced curve singularities over a Noetherian base scheme and prove that the delta invariant is semicontinuous. In our setting, each curve singularity in the family is the image of a parameterization and not the fiber of a morphism. The problem came up in connection with the right-left classification of parameterizations of curve singularities defined over a field of positive characteristic. We prove a bound for right-left determinacy of a parameterization in terms of delta and the semicontinuity theorem provides a simultaneous bound for the determinacy in a family. The fact that the base space can be an arbitrary Noetherian scheme causes some difficulties but is (not only) of interest for computational purposes.

Classification of Unimodal Parametric Plane Curve Singularities in Positive Characteristic

“…As before we may assume that a 12 = a 13 = a 14 = 0. We first prove that (t x,t4,y,t9+t10+t11+a*t15),15); poly x,t4,y,t9+t11+a*t15),15); W; 0 X; 0 Now we assume that a 15 = a 16 = a 17 = a 18 = 0. We have to prove that (t 4 , t 9 + t 11 + a 19 t 19 + .…”

“…The following determinacy bound and the semicontinuity of δ is proved in [15] for parametrizations of arbitrary (not necessay plane) reduced curve singularities and for arbitrary fields. Proposition 2.2 ([15], Proposition 11).…”

“…The theorem is proved in [15] more generally for A an arbitrary Noetherian ring and also for non-closed points in Spec(A).…”