On plane curves with several singular points with high multiplicity (Q1917221)

The authors consider some properties of families of smooth curves which are the normalizations of plane curves of degree \(n\) with \(s\) ordinary singular points of multiplicity \(m_1,..., m_s\) at fixed points \(P_1,...,P_s\) and with \(c\) nodes as other singularities. The family of such curves is denoted \(W(n,m_i,P_i,c)\) and the family of its normalizations \(NW(n,m_i,P_i,c)\). The authors study the set of nodes of curves in \(NW(n,m_i,P_i,c)\). Particularly, they show that for \(n\) large enough, in all its principal components (= components containing deformations of arrangements of \(n\) lines intersecting in the points \(P_i\) with multiplicities \(m_i\)), the monodromy of nodes is the full symmetric group, and there exists a component whose general curve has nodes of maximal rank. They also show the nonexistence of some linear series on curves in \(NW(n,m_i,P_i,c)\) under certain numerical constraints on the data.