The Groebner stratum of a monomial ideal is an affine variety that parameterizes the family of all ideals having 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 . Using properties of the Groebner strata we prove some sufficient conditions for the rationality of components of . We show for instance that all the smooth, irreducible components in (or in its support) and the Reeves and Stillman component 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.
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Paolo Lella, Margherita Roggero, Rational Components of Hilbert Schemes. Rend. Sem. Mat. Univ. Padova 126 (2011), pp. 11–45