[Paper Review] The space of Cohen-Macaulay curves

Katharina Heinrich|arXiv (Cornell University)|Mar 24, 2014

Algebraic Geometry and Number Theory15 references3 citations

TL;DR

This paper constructs a scheme $W$ with a representable, surjective, and smooth morphism $\pi: W \to \mathrm{CM}$, proving that the moduli functor $\mathrm{CM}$ parameterizing Cohen–Macaulay curves with finite maps to $\mathbb{P}^n$ that are generically closed immersions is an algebraic space. It further establishes that $\mathrm{CM}$ satisfies the valuative criterion for properness, confirming it is a proper algebraic space, using explicit geometric arguments based on coherent sheaves and depth conditions on the structure sheaf.

ABSTRACT

One can consider the Hilbert scheme as a natural compactification of the space of smooth projective curves with fixed Hilbert polynomial. Here we consider a different modular compactification, namely the functor CM parameterizing curves together with a finite map to $\mathbb{P}^n$ that is generically a closed immersion. We prove that CM is an algebraic space by contructing a scheme W and a representable, surjective and smooth map W -> CM. Moreover, we show that CM satisfies the valuative criterion for properness.

Motivation & Objective

• To re-prove that the moduli functor $\mathrm{CM}$ of Cohen–Macaulay curves with finite generically closed immersions into $\mathbb{P}^n$ is an algebraic space, addressing gaps in prior proofs.
• To provide a more explicit and simplified construction of a smooth, surjective, and representable scheme cover $\pi: W \to \mathrm{CM}$, replacing earlier incomplete arguments.
• To establish the properness of $\mathrm{CM}$ using a direct, geometric proof based on depth and coherence properties of structure sheaves, rather than abstract criteria.
• To clarify and strengthen the foundational framework for moduli spaces of curves that avoid degenerate subschemes present in the Hilbert scheme compactification.

Proposed method

• Construct a scheme $W$ parameterizing pairs $(C, i)$ with $C$ flat over a base and $i: C \to \mathbb{P}^n_S$ finite and generically a closed immersion, using explicit geometric data and coherent sheaf constructions.
• Define $\mathrm{CM}(S)$ as the set of isomorphism classes of such pairs satisfying purity, Cohen–Macaulay, and Hilbert polynomial conditions on fibers.
• Use the scheme-theoretic closure of the image in $\mathbb{P}^n_R$ over a discrete valuation ring $R$ to construct a limit of a generic family.
• Define a coherent $\mathcal{O}_Z$-algebra $\mathcal{A}$ as the pushforward of a gluing of $\mathcal{O}_{U_1}$ and $(h_K)_*\mathcal{O}_{C_K}$ over a dense open subset $U \subset Z$, and set $C := \mathbf{Spec}(\mathcal{A})$.
• Prove $\mathcal{A}$ is coherent and Cohen–Macaulay as an $\mathcal{O}_Z$-module using codimension and depth arguments, ensuring $C$ is flat and Cohen–Macaulay.
• Verify that the closed fiber $C_0$ is Cohen–Macaulay and $i_0$ is an isomorphism away from finitely many points by analyzing the non-Cohen–Macaulay locus and associated points.

Experimental results

Research questions

• RQ1 Is the moduli functor $\mathrm{CM}$ of Cohen–Macaulay curves with finite generically closed immersions into $\mathbb{P}^n$ representable by an algebraic space?
• RQ2 Can a smooth, surjective, and representable scheme cover $\pi: W \to \mathrm{CM}$ be explicitly constructed to replace earlier incomplete arguments?
• RQ3 Does $\mathrm{CM}$ satisfy the valuative criterion for properness, ensuring limits of families exist and are unique?
• RQ4 How can the coherence and Cohen–Macaulay property of the structure sheaf be used to ensure flatness and geometric finiteness in degeneration?

Key findings

• The moduli functor $\mathrm{CM}$ is an algebraic space, as shown by constructing a scheme $W$ and a representable, surjective, and smooth morphism $\pi: W \to \mathrm{CM}$.
• The construction of $W$ provides a more explicit and simplified alternative to earlier proofs, resolving gaps in Hønsen's original argument.
• The space $\mathrm{CM}$ satisfies the valuative criterion for properness, meaning every family over the generic point of a discrete valuation ring extends uniquely to the special fiber.
• The structure sheaf $\mathcal{A}$ of the total space $C$ is coherent and Cohen–Macaulay as an $\mathcal{O}_Z$-module, ensuring $C$ is flat and Cohen–Macaulay over $\operatorname{Spec}(R)$.
• The closed fiber $C_0$ is Cohen–Macaulay and the map $i_0: C_0 \to \mathbb{P}^n_k$ is an isomorphism away from finitely many points, satisfying the geometric condition for $\mathrm{CM}$.
• The Hilbert polynomial is preserved under base change, confirming that the limit object lies in $\mathrm{CM}(\operatorname{Spec}(R))$.