On the base space of a semi-universal deformation of rational quadruple points (Q1185442)

The authors study the deformation theory of a normal surface singularity \((X,0)\subset(\mathbb{C}^ n,0)\) by means of the projection method, developed by the authors in this article, in Abh. Math. Semin. Univ. Hamb. 60, 177- 208 (1990; Zbl 0721.32014) and in Math. Ann. 288, No. 3, 527-547 (1990; Zbl 0715.32013)]. The idea is to project \((X,0)\) generically into \((\mathbb{C}^ 3,0)\) and study the ``admissible'' deformation theory of the image \((\overline X,0)\) which is a weakly normal hypersurface with one- dimensional singular locus \(\Sigma\). The following theorem (which allows the choosing of very special projections) turns out to be a powerful tool in determining the base space \(Def(X)\) of the semi-universal deformation of \((X,0)\) up to a smooth factor: (i) Let \(Def(\overline X,\Sigma)\) be the base space of admissible deformations of \(\overline X\), then \(Def(\overline X,\Sigma)\cong Def(X)\times\mathbb{C}^ s\) for some \(s\geq 0\). (ii) Let \((\overline X_ 1,0)\) and \((\overline X_ 2,0)\) be two weakly normal surface singularities in \((\mathbb{C}^ 3,0)\) with defining equations \(f_ 1\) and \(f_ 2\) and (reduced) singular locus \(\Sigma_ 1\) and \(\Sigma_ 2\). Then \(Def(\overline X_ 1,\Sigma_ 1)\) and \(Def(\overline X_ 2,\Sigma_ 2)\) differ by a smooth factor if one of the following holds: -- \((\overline X_ 1,0)\) and \((\overline X_ 2,0)\) have the same normalization; -- \(\Sigma_ 1=\Sigma_ 2\) and \(f_ 1\equiv f_ 2\) modulo the square of the ideal defining \(\Sigma_ 1\). Using this, the authors prove: For \((X,x)\) rational and of multiplicity four, \(Def(X,0)\cong B(n)\times(\mathbb{C}^ s,0)\) where \(B(n)\) is a ``universal'' singularity depending only on one natural number \(n\) and which has the following properties: 1. The reduction of \(B(n)\) has \(n+1\) irreducible components \(Y_ 0,\dots,Y_ n\), \(\dim Y_ k=2n-1+2k\), 2. the normalizaton of \(Y_ k\) is smooth for all \(k\), 3. the multiplicity of \(Y_ k\) is \({n\choose k}\), 4. \(n\) is determined by the resolution graph of \((X,0)\). Moreover, the authors give equations for the space \(B(n)\) which are conceptually very simple.