Murphy's law in algebraic geometry: Badly-behaved deformation spaces
The author considers the following question: ``How bad can the deformation space of an object be?'' It is shown that usually the deformation space may be as bad as possible according to the philosophy ``There is no geometric possibility so horrible that it cannot be found generically on some component of some Hilbert scheme'' [cf. \textit{J. Harris, I. Morrison}, ``Moduli of Curves'', Grad. Texts Math. 187, Springer (1998; Zbl 0913.14005)]. By definition Murphy's law holds for a moduli space if every singularity type of finite type over \(\mathbb Z\) appears on that moduli space. It is proved that a lot of the well understood moduli spaces satisfy Murphy's law. For instance the Hilbert scheme of nonsingular curves in projective space, the Hilbert scheme of surfaces in \(\mathbb P^4\), the versal deformation spaces of isolated normal Cohen--Macaulay threefold singularities.
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