Deformations of the normalization of hypersurfaces

Abstract

Introduction A deformation theory has been introduced by Siersma [Si 11 for the simplest class of non-isolated singularities X: hypersurfaces with a smooth one dimensional singular locus ~ and transverse to a general point of E an AI singularity. He considered deformations, loosely speaking, such that Z stays inside the singular locus of the deformed X. This notion has been extended by Pellikaan [Pe 1] and has been studied further by the authors in [J-S 11 and [J-S 2]. Let us be more precise about the deformation functor we are interested in. Let C be a category of spaces (e.g. those of germs of analytic spaces). A diagram of spaces Z s ~ Xs, flat over some base space S, is called admissible iffZs ~ ~xsls, where CCxs/s is the relative critical space as defined by Teissier l-Te, p. 5871. Now let Z ~ X be an admissible diagram over the spectrum of the ground field. Then the functor of admissible deformations, Def(Z, X): C--*Set, is defined by: S~{isomorphism classes of deformations of ~ X over S which are admissible}. Of course, in general one expects this deformation functor to be obstructed. That this is also the case (in general) when X is a hypersurface in C 3, with a one dimensional reduced 2~, is shown by a beautiful example, due to Pellikaan. He considered the singularity X, defined by (yz) 2 + (xz) 2 + (xy) 2 = 0, with Z defined by the ideal (yz, xz, xy). He gave an admissible deformation over the space S defined by the equations ea = eb = ec = 0 in C 4, and showed that there are obstructions. This example has been worked out further in [J-S 1], IJ-S 2] and [J-S 3, Example 3.3]. The first example of an obstructed rational singularity has been given by Pinkham [Pil. He showed that the base space of a semi-universal deformation of the cone over the rational normal curve of degree 4 is isomorphic to the space S above. The nice thing is that the examples of Pellikaan and Pinkham are closely related: the normalization of Pellikaan's example is Pinkham's example! Or, to put it in another way, Pellikaan's example is a projection into •3 of Pinkham's example.