Equisingular Families of Projective Curves

Abstract

In this survey, we report on progress concerning families of projective curves with fixed number and fixed (topological or analytic) types of singularities. We are, in particular, interested in numerical, universal and asymptotically proper sufficient conditions to guarantee the nonemptyness, T-smoothness and irreducibility of the variety of all projective curves with prescribed singularities in a fixed linear system. We also discuss the analogous problem for hypersurfaces of arbitrary dimension with isolated singularities, and we close with a section on open problems and conjectures.

On rigid plane curves

“…It has a natural equisingular stratification with the strata determined by the collection of degrees and multiplicities of irreducible components and by the collection of topological singularity types of the considered curves (see [10,11]; below, the strata will be called the families of equisingular curves). Properties of this stratification have been studied by algebraic geometers since 19th century, attracting attention of leading experts like Zeuthen, Severi, Segre, Zariski and others (see, for example, [2] for a modern survey in this area).…”

“…[2]). It is, of course, invariant with respect to the action of Aut(P 2 ), and hence consists of entire orbits of the Aut(P 2 )-action.…”

“…To prove Theorem 3.1 we need in the following result. Then the family V n = V d n ; g; (n (1,2) + n (1,3) + n (1,4) ) A 2n−1 + · · · is strictly rigid for each n ≥ 2. The non-essential part of singularities of C ∈ V n consists of simple singularities.…”

“…Let L i , i = 1, 2, 3, be the tangent line to C at the point p i . It is easy to see that (1,2) + n (1,3) + n (1,4) …”

“…Under assumption that C has no other singularities in C 2 , the genus value follows from the fact that δ(T n,2n−1 ) = (n − 1) 2 …”