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On the regularity of the monomial point of a border basis scheme. (English) Zbl 1451.13081

An order ideal \(\mathcal O\) in the polynomial ring \(K[x_1,\dots,x_n]\) in \(n\) variables over a field \(K\) is a set of terms \(\sigma=x_1^{\alpha_1}\dots x_n^{\alpha_n}\) such that any term \(\tau\) dividing \(\sigma\) belongs to \(\mathcal O\). If \(\mathcal O\) is a finite order ideal, then the border basis scheme \(\mathbb B_{\mathcal O}\) is the affine scheme which parameterizes all the Artinian ideals \(I\) in \(K[x_1,\dots,x_n]\) with a border basis with respect to \(\mathcal O\) or, equivalently, such that \(\mathcal O\) is a basis of the quotient \(K[x_1,\dots,x_n]/I\) as a \(K\)-vector space. It is well-known that \(\mathbb B_{\mathcal O}\) can be embedded as an open subscheme in the Hilbert scheme of the \(0\)-dimensional schemes in \(\mathbb P^n_K\) with Hilbert polynomial \(p(z)=\vert \mathcal O \vert\), and that its defining ideal \(I(\mathbb B_{\mathcal O})\) is generated by quadratic polynomials.
The monomial ideal \(J\) that is generated by the terms outside \(\mathcal O\) corresponds to the origin of the affine space in which \(\mathbb B_{\mathcal O}\) is described as affine scheme. Thus, the dimension of the tangent space at \(J\) to \(\mathbb B_{\mathcal O}\), and hence to the Hilbert scheme, is computable from the generators of \(I(\mathbb B_{\mathcal O})\). More precisely, the cotangent space at \(J\) is computable. As a consequence, informations about the regularity of the border scheme and of the Hilbert scheme at \(J\) can be obtained. Thus, up to suitable changes of variables, the regularity of every point of the border basis scheme can be investigated.
The paper under review presents an efficient algorithm that computes a basis of the cotangent space at \(J\). Relying on previous papers of the first author with A. Kehrein or L. Robbiano, this algorithm arises from an analysis of the relations among the generators of the ideal \(J\) that directly provides the linear part of the quadratic generators of \(I(\mathbb B_{\mathcal O})\) (Proposition 2.7 and Corollary 2.8). The paper ends with a brief comparison with a method introduced by M. E. Huibregtse in an unpublished paper of 2005 [“The cotangent space at a monomial ideal of the Hilbert scheme of points of an affine space”, arXiv:math/0506575].
Other techniques and applications with analogous aims have been described in the setting of Gröbner schemes and in the term-order free setting of marked schemes over Pommaret bases, which give an open cover of Hilbert schemes with any Hilbert polynomial. For the Gröbner schemes, see [P. Lella and M. Roggero, Rend. Semin. Mat. Univ. Padova 126, 11–45 (2011; Zbl 1236.14006)], where a direct computation of the tangent space is provided. For the marked schemes, see section 1 of the paper [C. Bertone, F. Cioffi and M. Roggero, “Smoothable Gorenstein points via marked bases and double-generic initial ideals”, Exp. Math. (to appear), doi:10.1080/10586458.2019.1592034].

MSC:

13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14C05 Parametrization (Chow and Hilbert schemes)
14Q99 Computational aspects in algebraic geometry
13C05 Structure, classification theorems for modules and ideals in commutative rings

Citations:

Zbl 1236.14006

Software:

ApCoCoA
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References:

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