A genericity theorem for algebraic stacks and essential dimension of hypersurfaces

TL;DR

This paper determines the essential dimension of classes of homogeneous polynomials and hypersurfaces over fields of characteristic zero, introducing a new genericity theorem for algebraic stacks that has broader applications.

Contribution

It introduces a novel genericity theorem for algebraic stacks and applies it to compute the essential dimension of polynomial and hypersurface classes, expanding understanding of their classification.

Findings

Computed essential dimension of Forms_{n,d} and Hypersurf_{n,d}

Established a new genericity theorem for algebraic stacks

Applied the theorem to local complete intersection curves

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 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.