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Jason Starr
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If all you want is Cohen-Macaulay, this is straightforward, and your sketch proves the result. Every locally finite type scheme over a field has a maximal open subscheme that is Cohen-Macaulay (equivalent to local equidimensionality plus flatness of one / any quasifinite dominant morphism to a smooth scheme). By homogeneity, this open subscheme equals the entire group scheme.

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