Reducible Families of Curves with Ordinary Multiple Points on Surfaces in Projective Three-Space

TL;DR

This paper constructs examples of irreducible curves with ordinary multiple points on surfaces in projective three-space that are reducible and have components with unexpected dimensions, highlighting the limits of existing smoothness conditions.

Contribution

It provides new examples demonstrating the reducibility of certain families of curves with ordinary multiple points, challenging previous assumptions about T-smoothness conditions.

Findings

Examples of reducible families with ordinary multiple points

At least one component does not have the expected dimension

Conditions for T-smoothness are asymptotically accurate

Abstract

In math.AG/0108089, math.AG/0212090 and math.AG/0308247 we gave numerical conditions which ensure that an equisingular family is irreducible respectively T-smooth. Combining results by Greuel, Lossen and Shustin and an idea from math.AG/9802009 we give in the present paper series of examples of families of irreducible curves on surfaces in projective three-space with only ordinary multiple points which are reducible and where at least one component does not have the expected dimension. The examples show that for families of curves with ordinary multiple points the conditions for T-smoothness in math.AG/0308247 have the right asymptotics.