JournalsrsmupVol. 126pp. 11–45

Rational Components of Hilbert Schemes

  • Paolo Lella

    Università degli Studi di Torino, Italy
  • Margherita Roggero

    Università degli Studi di Torino, Italy
Rational Components of Hilbert Schemes cover
Download PDF


The Groebner stratum of a monomial ideal j\mathfrak{j} is an affine variety that parameterizes the family of all ideals having j\mathfrak{j} as initial ideal (with respect to a fixed term ordering). The Groebner strata can be equipped in a natural way with a structure of homogeneous variety and are in a close connection with Hilbert schemes of subschemes in the projective space Pn\mathbf{P}^n. Using properties of the Groebner strata we prove some sufficient conditions for the rationality of components of Hilbp(z)n{\mathcal{H}\text{ilb}_{p(z)}^n}. We show for instance that all the smooth, irreducible components in Hilbp(z)n{\mathcal{H}\text{ilb}_{p(z)}^n} (or in its support) and the Reeves and Stillman component HRSH_{RS} are rational. We also obtain sufficient conditions for isomorphisms between strata corresponding to pairs of ideals defining a same subscheme, that can strongly improve an explicit computation of their equations.

Cite this article

Paolo Lella, Margherita Roggero, Rational Components of Hilbert Schemes. Rend. Sem. Mat. Univ. Padova 126 (2011), pp. 11–45

DOI 10.4171/RSMUP/126-2