Specializations of complex quartic curves

Extract

Let F: ⁿ × ^r → be a sufficiently small representative, defined near (0,0), of a versal unfolding of a germ F₀: (ⁿ, 0) → (, 0) with isolated singularity at 0∈ⁿ. Then a result of Teissier ((4), p. 338) says that, for u∈^r sufficiently close to 0, F acts as a versal unfolding of all the various singularities, close to 0, on the fibre F_u = 0 (where F_u(x) = F(x, u)). Let us fix a small neighbourhood W of 0 in ⁿ and restrict u to be so close to 0 that all singularities of F_u lie in W. Suppose that the fibre F_u = 0 has singularities of (contact) type χ₁, …, χ_m, all isolated. Suppose that, for each i, the collection {χ_ij}(1 ≤ j ≤ k(i)) of singularities specializes to χ_i, that is it occurs on a single fibre arbitrarily close to χ_i in a versal unfolding of χ_i. Then Teissier's result shows that there will exist fibres F_v = 0 of F, with v arbitrarily close to u, such that the fibre F_v = 0 carries singularities, all in W, of all the types χ_ij for 1 ≤ i ≤ m, 1 ≤ j ≤ k(i). In other words, if ‘despecializations’ {χ_ij} of the separate singularities χ_i are possible, then they can all occur together on a single fibre of F, provided F is versal.