The algebra of glued polynomials on an arrangement of hyperplanes
Let \(F\) be an infinite field and \({\mathcal A} = \{H_ 1, \dots, H_ n\}\) a nonempty set (arrangement) of hyperplanes in an \(F\)-vector space \(V\) of dimension \(l\). Let \(S\) be the symmetric algebra of \(V^*\). Associated to \({\mathcal A}\) are two graded algebras which are graded \(S\)-modules: The coordinate ring \(R\) of the affine variety \(\bigcup^ n_{i=1} H_ i\), the second one \(^ +R\) is the pull-back of the system of the coordinate rings \(R(X)\) of various intersections \(X\) of the hyperplanes and the natural projections \(\rho_{X,Y} : R(X) \to R(Y)\). In fact \(^ +R\) is the seminormalisation of \(R\). The study of the Hilbert-Poincaré series of \(^ +R\) constitutes a central point in this paper. As a consequence the authors give examples of arrangements with non Cohen-Macaulay \(^ +R\) (of course \(R\) is Cohen-Macaulay). They suggest that all arrangements of a given dimension and of a given number of hyperplanes can be broken in several nontrivial classes depending on the depth of \(^ +R\).
scientific article; zbMATH DE number 681811