The versal deformation of universal curve singularities
Universal curve singularities, or partition curves, are defined by the property that the conductor ideal is equal to the maximal ideal in the local ring of the singularity. The simplest examples are the curves \(L^m_m\) consisting of the union of the \(m\) coordinate axes in \(\mathbb{C}^m\). For each partition of the multiplicity \(m\) into \(r\) numbers \(m_i\) there exists a unique universal curve singularity with \(r\) branches, of multiplicities \(m_i\). Such a branch has parametrization \((t^{m_i}, t^{m_1 + 1}, \dots, t^{2m_i- 1})\). The base space of a versal deformation of \(L^m_m\) is given by the system of equations \[ (a_{ik} - a_{ij}) (a_{jk} - a_{ji}) = (a_{il} - a_{ij}) (a_{jl} - a_{ji}) \] on a set of \(m(m - 1)\) unknowns \(a_{ij}\). For \(m \leq 3\) this variety is smooth, for \(m = 4\) it is isomorphic to the cone over the Segre embedding of \(\mathbb{P}^1 \times \mathbb{P}^3\), for \(m = 5\) it is the cone over a variety \(V\) which admits \(\mathbb{P}^4 \times P\), where \(P\) is a Del Pezzo surface of degree 5 as small resolution. The author describes the structure of the space for general \(m\) as a reciprocally generated variety. To obtain results for other partition curves, the author uses the so-called projection method of \textit{T. de Jong} and the reviewer. He obtains the result that for all partition curves of multiplicity 4 the base space is, up to a smooth factor, isomorphic to the base spaces of \(L^4_4\), but for multiplicity 5 there is no analogous result and the structure of the base space depends on the particular partition of \(m\), contrary to an earlier claim of A. G. Aleksandrov. The paper contains a huge amount of equations that were generated by the computer program MACAULAY.
scientific article; zbMATH DE number 492593
scientific article