A genericity theorem for algebraic stacks and essential dimension of hypersurfaces

Abstract

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 ${\mathbb{P}}^{n-1}$, respectively, over any base field $k$ of characteristic $0$. Here two polynomials (or hypersurfaces) over $K$ 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.