The space of Cohen–Macaulay curves and related topics

The space of Cohen–Macaulay curves is a compactification of the space of curves that are embedded in a given projective space Pn. The idea is similar to that of the Hilbert scheme but instead of adding degenerated curves, one considers only curves without embedded or isolated points. However, the curves need not be embedded into the projective space. Instead, they come with a finite morphism to Pn that is generically a closed immersion. More precisely, the space CM of Cohen–Macaulay curves parameterizes flat families of pairs (C, i) where C is a curve without embedded or isolated points and i:C ! Pn is a finite morphism that is an isomorphism onto its image away from finitely many closed points and such that C has Hilbert polynomial p(t) with respect to the map i. In Paper A we show that the moduli functor CM is represented by a proper algebraic space. This is done by constructing a smooth, surjective cover ⇡:W ! CM and by verifying the valuative criterion for properness. Paper B studies the moduli space CM in the particular case n = 3 and p(t) = 3t+1, that is, the Cohen–Macaulay space of twisted cubics. We describe the points of CM and show that they are in bijection with the points on the 12-dimensional component H0 of the Hilbert scheme of twisted cubics. Knowing the points of CM , we can then show that the moduli space is irreducible, smooth and has dimension 12. Paper C concerns the notion of biequidimensionality of topological spaces and Noetherian schemes. In EGA it is claimed that a topological space X is equidimensional, equicodimensional and catenary if and only if all maximal chains of irreducible closed subsets in X have the same length. We construct examples of topological spaces and Noetherian schemes showing that the second property is strictly stronger. This gives rise to two different notions of biequidimensionality, and we show how they relate to the dimension formula and the existence of a codimension function.