A genericity theorem for algebraic stacks and essential dimension of hypersurfaces

作者：Reichstein Zinovy^*; Vistoli Angelo

来源：Journal of the European Mathematical Society, 2013, 15(6): 1999-2026.

DOI：10.4171/JEMS/411

• Essential dimension
• hypersurface
• genericity theorem
• stack
• gerbe

We compute the essential dimension of the functors Forms(n,d) and Hypersurf(n,d) of equivalence classes of homogeneous polynomials in n variables and hypersurfaces in Pn-1, respectively, over any base field k of characteristic 0. Here two polynomials (or hypersurfaces) over a field K / k are considered equivalent if they are related by a linear change of coordinates with coefficients in K. 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.

• 出版日期2013