Macaulay-like marked bases

Abstract

We define marked sets and bases over a quasi-stable ideal j in a polynomial ring on a\ud Noetherian K-algebra, with K a field of any characteristic. The involved polynomials\ud may be non-homogeneous, but their degree is bounded from above by the maximum\ud among the degrees of the terms in the Pommaret basis of j and a given integer m.\ud Due to the combinatorial properties of quasi-stable ideals, these bases behave well with\ud respect to homogenization, similarly to Macaulay bases. We prove that the family of\ud marked bases over a given quasi-stable ideal has an affine scheme structure, is flat and,\ud for large enough m, is an open subset of a Hilbert scheme. Our main results lead to\ud algorithms that explicitly construct such a family. We compare our method with similar\ud ones and give some complexity results

Smoothable zero dimensional schemes and special projections of algebraic varieties

“…The isomorphism is given by sending ( , ) to the irreducible scheme Sym ∨ ∕( ) + ( ∨ ) 3 translated so that its support is . Hence the schemes corresponding to points of Hilb 143 are irreducible and have Hilbert function (1,4,3). By apolarity we may equivalently parameterise this component as × Gr ( 3, Sym 2 ) , by sending ( , ) to Apolar( + ) supported at .…”

“…Hence we may consider only * -limits. Let = lim →0 Γ be a * -limit in 4 with Hilbert function (1,4,3). Compactify 4 to ℙ 4 and consider Γ ⊂ ℙ 4 .…”

“…Then is smoothable if and only if the corresponding point lies in . Whether a given R is smoothable is a difficult question, see . It is connected with the search for equations of secant varieties .…”

“…(a) • 2 ∶ ℙ 3 → ℙ 6 is a restriction of a Cremona transformation ℙ 6 ⤏ ℙ 6 of type (2,4) based in a rational octic surface or a limit of such restrictions, (b) is contained in a cubic hypersurface, (c) is contained in a three-dimensional space of cubic hypersurfaces, (d) the scheme = Spec Apolar( ) is smoothable, (e) is spanned by partial derivatives of a cubic form ∈ Sym 3 .…”

“…In particular, Cremona transformations with base loci being smooth surfaces was classified in [14]. In this work we investigate and exploit the geometry of one of the five types of transformations with base loci being smooth surfaces: Cremona transformations of type (2,4) in ℙ 6 based in a rational surface embedded as a surface of degree 8 and sectional genus 3 in ℙ 6 .…”