Chow-stability and Hilbert-stability in Mumford's Geometric Invariant Theory

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In this note, we shall show that the Chow-stability and the Hilbert-stability in GIT asymptotically coincide.
I slightly changed the title of the paper. Moreover, a proof using an equivariant version of Serre's Conjecture is given. This new version appeared in Osaka J. Math. 45 (2008)

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