[Paper Review] Stability of Tails and 4-Canonical Models

Donghoon Hyeon, Ian Morrison|ArXiv.org|Jun 9, 2008

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper establishes that the GIT quotient of the 4-canonical Hilbert and Chow loci of genus $g \geq 3$ curves yields the moduli space $\overline{M}_g^{\text{ps}}$ of pseudostable curves. By computing exact weights of one-parameter subgroups acting on $m$-th Hilbert points of curves with elliptic or cuspidal tails, the authors prove that such curves are $m$-Hilbert unstable and Chow strictly semistable, confirming that the quotient parametrizes pseudostable curves with no elliptic tails and only nodes and ordinary cusps.

ABSTRACT

We show that the GIT quotients of suitable loci in the Hilbert and Chow schemes of 4-canonically embedded curves of genus $g\ge 3$ are the moduli space $\bar{M}_g^{ ext{ps}}$ of pseudo-stable curves constructed by Schubert in \cite{Schubert} using Chow varieties and 3-canonical models. The only new ingredient needed in the Hilbert scheme variant is a more careful analysis of the stability with respect to a certain 1-ps $λ$ of the $m^{ ext{th}}$ Hilbert points of curves $X$ with elliptic tails. We compute the exact weight with which $λ$ acts, and not just the leading term in $m$ of this weight. A similar analysis of stability of curves with rational cuspidal tails allows us to determine the stable and semistable 4-canonical Chow loci. Although here the geometry of the quotient is more complicated because there are strictly semi-stable orbits, we are able to again identify it as $\bar{M}_g^{ ext{ps}}$. Our computations yield, as byproducts, examples of both $m$-Hilbert unstable and $m$-Hilbert stable $X$ that are Chow strictly semi-stable.

Motivation & Objective

• To complete the GIT construction of moduli spaces for $\nu$-canonical curves by resolving the $\nu=4$ case.
• To show that the $4$-canonical Hilbert and Chow quotients parametrize pseudostable curves, excluding elliptic tails.
• To compute exact weights of one-parameter subgroups acting on $m$-th Hilbert points of curves with elliptic or cuspidal tails.
• To clarify the relationship between Hilbert and Chow stability in the $4$-canonical setting, particularly for strictly semistable loci.
• To confirm that $\overline{M}_g^{\text{ps}}$ arises as the GIT quotient of both the Hilbert and Chow schemes for $4$-canonical embeddings.

Proposed method

• Precisely compute the $\rho^{-1}$-weight of the $m$-th Hilbert point for curves with elliptic tails using explicit monomial bases and order of vanishing at singularities.
• Analyze the action of a $1$-ps $\rho$ on the $m$-th Hilbert point of $4$-canonical models of curves with elliptic tails, computing the exact weight $\mu([Z]_m, \rho^{-1}) = m-1$ for cusp curves.
• Use the exact weight computation to show that $m$-th Hilbert points of $4$-canonical models of curves with elliptic tails are $\rho$-unstable, extending Schubert’s $3$-canonical result.
• Apply the same method to curves with rational cuspidal tails, showing they are $m$-Hilbert stable and Chow strictly semistable for $m \geq 2$, with $\mu([Z]_m, \rho^{-1}) = m-1$.
• Establish that the $4$-canonical Hilbert and Chow quotients are isomorphic and isomorphic to $\overline{M}_g^{\text{ps}}$ via injectivity, birationality, and normality arguments.
• Use Kollár’s results on flat universal families and regularity of the inverse to the fundamental cycle map to prove that the Hilbert and Chow quotients are isomorphic.

Experimental results

Research questions

• RQ1 Does the $4$-canonical Hilbert quotient of genus $g \geq 3$ curves yield the moduli space $\overline{M}_g^{\text{ps}}$ of pseudostable curves?
• RQ2 Are $m$-th Hilbert points of $4$-canonical models of curves with elliptic tails unstable under the action of a specific $1$-ps $\rho$?
• RQ3 What is the exact weight of the $1$-ps $\rho^{-1}$ acting on the $m$-th Hilbert point of a $4$-canonical curve with an ordinary cusp?
• RQ4 How do the semistable loci in the $4$-canonical Hilbert and Chow schemes relate to the moduli of weakly pseudostable and pseudostable curves?
• RQ5 Is the natural map $\varpi: \text{Hilb}_{g,4}/\!\!/SL_{7g-7} \to \text{Chow}_{g,4}/\!\!/SL_{7g-7}$ an isomorphism?

Key findings

• The $m$-th Hilbert point of a $4$-canonical model of a curve with an elliptic tail is $\rho$-unstable, as shown by exact weight computation in Lemma 1 and Corollary 2.
• For $4$-canonical curves with an ordinary cusp, the $m$-th Hilbert point is $m$-Hilbert stable and the Chow point is Chow strictly semistable, with $\mu([Z]_m, \rho^{-1}) = m - 1$.
• The $4$-canonical Hilbert quotient $\text{Hilb}_{g,4}/\!\!/SL_{7g-7}$ is isomorphic to $\overline{M}_g^{\text{ps}}$, with the stable locus parameterizing pseudostable curves and no strictly semistable points.
• The $4$-canonical Chow quotient $\text{Chow}_{g,4}/\!\!/SL_{7g-7}$ is isomorphic to $\overline{M}_g^{\text{ps}}$, with the semistable locus parameterizing weakly pseudostable curves and identification occurring precisely for curves with the same Deligne-Mumford stabilization and either a cusp or an elliptic tail.
• The map $\varpi: \text{Hilb}_{g,4}/\!\!/SL_{7g-7} \to \text{Chow}_{g,4}/\!\!/SL_{7g-7}$ is an isomorphism, established via injectivity, birationality, and normality of both spaces.