| We address the problem of parameterizing reduced curves with given singularities. Associated to a reduced curve C over an algebraically closed field is a smooth curve D, namely the normalization of C, together with a finite closed subscheme Sigma of D, namely the reduced preimage of the non-smooth locus of C. If the only singularities of C are ordinary nodes (xy = 0) and ordinary cusps (y 2 = x3), then C can be recovered from the pair D, Sigma without further information by "crimping" D along Sigma, i.e. by forming the push-out of a suitable diagram of schemes. However, if C has more complicated singularities, it is necessary to specify a "crimping datum" along with each singularity in order to recover C from D, Sigma.;In Part 1, we show that there is a smooth moduli scheme parameterizing crimping data for a curve singularity over a field, which we call the crimping scheme of the singularity, and we describe its structure. More generally, we show how to make sense of crimping data over an arbitrary affine base scheme. We apply our results to describe moduli stacks of pointed reduced curves with given singularities. In Part 2, we work out the corresponding infinitesimal theory: the locally trivial deformation theory of a reduced curve and its marked normalization map. |
