Local properties of families of plane curves (Q1324787)

Let \(\mathbb{P}^ N\) be the projective space parametrizing projective plane curves of degree \(n\). Let \[ \Sigma_{n,d} \subset \mathbb{P}^ N \times \text{Sym}^ d(\mathbb{P}^ 2) \] be the closure of the locus of pairs \((E,\sum P_ i)\), where \(E\) is an irreducible nodal curve and \(P_ 1,\dots,P_ d\) are its nodes. Main theorem: \(\Sigma_{n,d}\) is unibranch everywhere. Let \(V(n,g)\subset \mathbb{P}^ N\) be the locus of reduced and irreducible curves of genus \(g\), where \[ g=(n-1)(n-2)/2-d. \] The theorem implies that \(V(n,g)\) is irreducible [``Harris' theorem'', cf. \textit{J. Harris}, Invent. Math. 84, 445-461 (1986; Zbl 0596.14017)] and unibranch everywhere.