Abstract
In this paper, we introduce the notion of a complete hypertetrahedral arrangement {mathcal {A}} in {mathbb {P}}^{n}. We address two basic problems. First, we describe the local freeness of {mathcal {A}} in terms of smaller complete hypertetrahedral arrangements and graph theory properties, specializing the Mustaţă–Schenck criterion. As an application, we obtain that general complete hypertetrahedral arrangements are not locally free. In the second part of this paper, we bound the initial degree of the first syzygy module of the Jacobian ideal of {mathcal {A}}.
Highlights
The study of the module Der(− log A) of logarithmic vector fields tangent to the reduced divisor DA of a hyperplane arrangement A = {H1, . . . , Hm} began with Saito in [12] and
For n = 2, hypertetrahedral arrangements coincide with the family of triangular arrangements introduced in [9]
In [12] it was proved that for any hyperplane arrangement A, its module of derivations Der(− log A) is reflexive. This grants the freeness of hyperplane arrangements in P1, and the local freeness of line arrangements in P2
Summary
Definition 2.2 A hyperplane arrangement A in Pn is free if Der(− log A) is a free R-module of rank n + 1. In this case, the degrees 1, d1, . Theorem 2.3 The hyperplane arrangement A is free if and only if there exist n+1 logarithmic derivations n θi = fi j ∂xi ∈ Der(− log A). (iii) The Jacobian ideal JA of the line arrangement A in P2 with defining equation fA = x0x1x2(x0 + x1 + x2) has a minimal free R-resolution:. Remark 2.6 In terms of the defining equations of the hyperplanes of a graphic arrangement A , we can characterize them as follows:. Gs θ ∈ (TA )|U and this concludes the proof
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