1104.2007Apr 11, 2011Added equations of the Hilbert schemes of 2 points in the plane, 3-space, 4-space and of 3 points in the plane (a Macaulay2 file with the complete computation is available at http://tinyurl.com/EquationsHilbPoints-m2). Final version. To appear on Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
Extensors and the Hilbert scheme
Jerome Brachat, Paolo Lella, Bernard Mourrain, Margherita Roggero
The Hilbert scheme Hilbp(t)n parametrizes closed subschemes and families of closed subschemes in the projective space Pn with a fixed Hilbert polynomial p(t)
Nearby in the stack
. It is classically realized as a closed subscheme of a Grassmannian or a product of Grassmannians. In this paper we consider schemes over a field
k
of characteristic zero and we present a new proof of the existence of the Hilbert scheme as a subscheme of the Grassmannian
Grp(r)N(r)
, where
N(r)=h0(OPn(r))
. Moreover, we exhibit explicit equations defining it in the Plücker coordinates of the Plücker embedding of
Grp(r)N(r)
. Our proof of existence does not need some of the classical tools used in previous proofs, as flattening stratifications and Gotzmann's Persistence Theorem. The degree of our equations is
degp(t)+2
, lower than the degree of the equations given by Iarrobino and Kleiman in 1999 and also lower (except for the case of hypersurfaces) than the degree of those proved by Haiman and Sturmfels in 2004 after Bayer's conjecture in 1982. The novelty of our approach mainly relies on the deeper attention to the intrinsic symmetries of the Hilbert scheme and on some results about Grassmannian based on the notion of extensors.
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