Some obstructed equisingular families of curves on surfaces in projective three-space

TL;DR

This paper constructs examples of obstructed equisingular families of irreducible curves with simple singularities on surfaces in projective three-space, highlighting cases where these families are not T-smooth and comparing conditions for smoothness.

Contribution

It provides new examples of obstructed equisingular families of curves on three-dimensional projective surfaces, extending known results beyond the projective plane.

Findings

Examples of non T-smooth families are constructed.

Conditions for the existence of T-smooth components are analyzed.

Asymptotic behavior of these conditions is compared.

Abstract

Very few examples of obstructed equsingular families of curves on surfaces other than the projective plane are known. Combining results from Westenberger and Hirano with an idea from math.AG/9802009 we give in the present paper series of examples of families of irreducible curves with simple singularities on surfaces in projective three-space which are not T--smooth, i.e. do not have the expected dimension, and we compare this with conditions (showing the same asymptotics) which ensure the existence of a T--smooth component.