JournalsjemsVol. 15, No. 6pp. 1999–2026

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
A genericity theorem for algebraic stacks and essential dimension of hypersurfaces cover
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Abstract

We compute the essential dimension of the functors Formsn,d_{n,d} and Hypersurfn,d_{n, d} of equivalence classes of homogeneous polynomials in nn variables and hypersurfaces in Pn1\mathbb P^{n-1}, respectively, over any base field kk of characteristic 00. Here two polynomials (or hypersurfaces) over KK are considered equivalent if they are related by a linear change of coordinates with coefficients in KK. 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