Mathematical Research Letters

Volume 17 (2010)

Number 4

Stability of Tails and $4$-Canonical Models

Pages: 721 – 729



Donghoon Hyeon (Postech, Pohang, Gyungbuk 790-784, Republic of Korea)

Ian Morrison (Mathematical Sciences Research Institute, Berkeley CA)


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 $\overline{M}_g^{\text{ps}}$ of pseudostable 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 $\lambda$ of the $m^{\text{th}}$ Hilbert points of curves $X$ with elliptic tails. We compute the exact weight with which $\lambda$ 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 semistable orbits, we are able to again identify it as $\overline{M}_g^{\text{ps}}$. Our computations yield, as byproducts, examples of both $m$-Hilbert unstable and $m$-Hilbert stable $X$ that are Chow strictly semistable.

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