Zinovy Reichstein and Angelo Vistoli: A genericity theorem for algebraic stacks and essential dimension of hypersurfaces

reichst@math.ubc.ca and angelo.vistoli@sns.it

Submission: 2011, Mar 8, revised 2011, Mar 19

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.

2010 Mathematics Subject Classification: Primary 14A20, 14J70

Keywords and Phrases: Essential dimension, hypersurface, genericity theorem, stack, gerbe

Full text: dvi.gz 61 k, dvi 139 k, ps.gz 451 k, pdf.gz 235 k, pdf 258 k.

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