G. Berhuy, Z. Reichstein, On the notion of canonical dimension for algebraic groups, Advances in Mathematics, 198 (2005), 128--171.

Abstract: We define and study a new numerical invariant of an algebraic group action; we call it the canonical dimension. We then apply the resulting theory to the problem of computing the minimal number of parameters required to define a generic hypersurface of degree d in P^{n-1}.

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