Cayley–Bacharach and Singularities of Foliations

Abstract

This paper deals with foliations by curves [s] of degree r ≥ 2 on P 2 with isolated singularities S, called non-degenerate if S is reduced and otherwise degenerate. Say that [s] is uniquely determined by a zero-dimensional Y ⊂ S if [s] is the unique foliation that vanishes on Y and say that Y is minimal for [s] if, moreover, the degree of Y is the minimal possible to do so. Previous work of the authors show that every nondegenerate foliation in degrees 2 ≤ r ≤ 5 does have a minimal subscheme and that the set of non-degenerate foliations of degree r ≥ 6 that have a minimal subscheme contains a Zariski-open subset of the space of foliations of degree r. For non-degenerate foliations [s] we present both characterizations and sufficient conditions for [s] to have a minimal subscheme. We also give examples of degenerate foliations of degree r ≥ 7 that do not have a minimal subscheme at all.