# A genericity theorem for algebraic stacks and essential dimension of hypersurfaces

### Zinovy Reichstein

University of British Columbia, Vancouver, Canada### Angelo Vistoli

Scuola Normale Superiore, Pisa, Italy

## 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.

## 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. 1999–2026

DOI 10.4171/JEMS/411