We compute the essential dimension of the functors Forms and Hypersurf of equivalence classes of homogeneous polynomials in variables and hypersurfaces in , respectively, over any base field of characteristic . Here two polynomials (or hypersurfaces) over are considered equivalent if they are related by a linear change of coordinates with coefficients in . Our proof is based on a new Genericity Theorem for algebraic stacks, which is of independent interest. As another application of the Genericity Theorem, we prove a new result on the essential dimension of the stack of (not necessarily smooth) local complete intersection curves.
Cite this article
Zinovy Reichstein, Angelo Vistoli, A genericity theorem for algebraic stacks and essential dimension of hypersurfaces. J. Eur. Math. Soc. 15 (2013), no. 6, pp. 1999–2026DOI 10.4171/JEMS/411