Low degree equations defining the Hilbert scheme

Details Low degree equations defining the Hilbert scheme Low degree equations defining the Hilbert scheme

2011-04-11

arxiv.org/abs/1104.2007v1

Mathematics

Algebraic Geometry

20 pages. Comments are welcome

Scientific paper

The Hilbert scheme $\mathcal{H}\textnormal{ilb}_{p(t)}^{n}$, parametrizing closed subschemes in the projective space $\mathbb{P}^n$ with Hilbert polynomial $p(t)$ of degree $d$, is classically constructed as a subscheme of a suitable grassmannian, so that it becomes a closed projective subscheme through the associated Pl\"ucker embedding. In this paper we find new global equations for $\mathcal{H}\textnormal{ilb}_{p(t)}^{n}$ and especially we prove that $\mathcal{H}\textnormal{ilb}_{p(t)}^{n}$ can be defined scheme-theoretically by homogeneous polynomials of degree smaller than or equal to $d+2$ in the Pl\"ucker coordinates. We obtain these equations using a special set of generators of any exterior power of a vector space, that depend linearly on the Pl\"ucker coordinates of such vector space in the corresponding grassmannian. Firstly, we exploit these generators to obtain in a simpler way the well-known equations for $\mathcal{H}\textnormal{ilb}_{p(t)}^{n}$ by Iarrobino and Kleiman and those conjectured by Bayer in 1982 and then proved by Haiman and Sturmfels in 2004. Finally, using combinatorial properties of Borel-fixed ideals and their relations with the geometry of the Hilbert scheme, we obtain our new equations. A procedure for computing this set of equations comes out directly of our proof and it can be fruitfully used in simple cases.

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