A Moduli Scheme of Embedded Curve Singularities

A central problem in Algebraic Geometry is the classification of several isomorphism classes of objects by considering their deformations and studying the naturally related moduli problems, see [33], [34]. This general strategy has also been applied to singularities. Some classes of singularities with fixed numerical invariants are studied from the moduli point of view, i.e. proving the existence of moduli spaces or giving obstructions to their existence. See for instance [17], [28], [29] and [42]. The main purpose of this paper is to prove the existence of the moduli space HN,p parameterizing the embedded curve singularities of (k, 0) with an admissible Hilbert polynomial p and to study its basic properties. The main difference between the classical projective moduli problems and the case studied here is that HN,p is not a locally finite type scheme. Hence the general techniques of construction of moduli spaces of projective objects do not apply to our problem and we need to develop specific ones. Since HN,p is a projective limit of k-schemes of finite type we define a measure μp in HN,p valued in the completion M̂ of the ring M = K0(Sch)[L] where L is the class of K0(Sch) defined by the affine line over k. This measure induces a motivic integration on HN,p and enable us to consider a motivic volume for singularities of arbitrary dimension. See [27], [7], and [30] for the motivic integration on jet schemes. In [11], see also [14], we characterized the Hilbert-Samuel polynomials of curve singularities: we proved that there exists a curve singularity C with embedding dimension b and Hilbert polynomial p = e0T − e1 if, and only if, either b = e0 = 1 and e1 = 0, or 2 ≤ b ≤ e0, and ρ0,b,e0 ≤ e1 ≤ ρ1,b,e0, see Theorem 3.1 for the definitions of ρ0,b,e0 and ρ1,b,e0 . Moreover, for each triplet (b, e0, e1) satisfying the above conditions there is a reduced curve singularity C ⊂ (k , 0) with k an algebraically closed field. From this result and the main result of this