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In this paper we study the Hilbert function of gr m (R), when R is a numerical semigroup ring or, equivalently, the coordinate ring of a monomial curve. In particular, we prove a sufficient condition for a numerical semigroup ring in order get a non-decreasing Hilbert function, without making any assumption on its embedding dimension; moreover, we show how this new condition allows to improve known results about this problem. To this aim we use certain invariants of the semigroup, with particular regard to its Apéry-set.

We show that in arithmetically-Gorenstein line arrangements with only planar singularities, each line intersects the same number of other lines. This number has an algebraic interpretation: it is the Castelnuovo-Mumford regularity of the coordinate ring of the arrangement.We also prove that every (d − 1)-dimensional simplicial complex whose 0-th and 1-st homologies are trivial is the nerve complex of a suitable d-dimensional standard graded algebra of depth ≥ 3. This provides the converse of a recent result

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