In a recent paper, Mezzetti, Miró‐Roig and Ottaviani [Mezzetti et al., ‘Laplace equations and the weak Lefschetz property’, Canad. J. Math. 65 (2013) 634–654] highlight the link between rational varieties satisfying a Laplace equation and artinian ideals failing the weak Lefschetz property. Continuing their work, we extend this link to the more general situation of artinian ideals failing the strong Lefschetz property. We characterize the failure of the SLP (which includes WLP) by the existence of special singular hypersurfaces (cones for WLP). This characterization allows us to solve three problems posed in J. C. Migliore and U. Nagel [‘A tour of the weak and strong Lefschetz properties’, Preprint, 2011, arXiv:1109.5718, September 2011. J Commutative Algebra, to appear] and to give new examples of ideals failing the SLP. Finally, line arrangements are related to artinian ideals and the unstability of the associated derivation bundle is linked to the failure of the SLP. Moreover, we reformulate the so‐called Terao's conjecture for free line arrangements in terms of artinian ideals failing the SLP.

Using liaison, we study projective curves which are close to complete intersections in terms of Castelnuovo-Mumford regularity or degree and we obtain conditions for a space curve C being arithmetically Cohen-Macaulay, generalizing a result of E. D. Davis. These conditions apply also, in weaker forms, to equidimensional and lCM subschemes of codimension two and to curves in P n , with n ≥ 4. The results for curves are sharp as suitable examples show.Keywords Cohen-Macaulayness · Liaison · Hilbert function · Castelnuovo-Mumford regularity IntroductionApplying liaison techniques, we study projective curves C contained in complete intersections Y under conditions on the difference reg(Y )−reg(C) between Castelnuovo-Mumford regularities or on the difference deg(Y ) − deg(C) between degrees. By a projective curve we mean an equidimensional projective subscheme of dimension 1, hence a locally CohenMacaulay (lCM for short) projective subscheme of dimension 1.To address the question, let C ⊂ P n K be a curve, Y a complete intersection of type (β 1 , . . . , β n−1 ) containing C, C the curve algebraically linked to C by Y and ρ C the regularity of the Hilbert function of C. In [3] it is proved that, if C is a space curve such that reg(C) = reg(Y ) − 1 > ρ C + 1, then C is a plane curve, hence arithmetically CohenMacaulay (aCM for short). Starting from this result, in Sect. 3 we investigate curves C ⊂ P n K

In this note we generalize the main result in the previous work on JLMS on artinian ideals failing Lefschetz properties, varieties satisfying Laplace equations and existence of suitable singular hypersurfaces. Moreover we characterize the minimally generation of ideals generated by power of linear forms by the conguration of their dual points in the projective plane. Finally we show the equivalence among failing SLP, Laplace equations and unexpected curves

We study cun •es 011 a smooth ratio11al scroll surface S, i11 pm·ticular the multiplicath•e structure of the Hartslwme-Rao m{){/ule l\1 c of a11y cun•e C c S . The mai11 re. m lt is the co11structio11 of the mi11imal !(ellemtors of M c · As a comeque11ce, we get that for mrves C 011 a ratio11al 11ormal scroll smfuce, the Hilbert fu11 ctio11 of M c determi11es the module structure. This is a strOll/( form of the com•erse of the Har tslwme-Scltell:el Theoretn.

In this paper, using the method of moving frames, we generalise some of Terracini's results on varieties with tangent defect. In particular, we characterise varieties with higher order osculating defect in terms of Jacobians of higher fundamental forms and moreover we characterise varieties with "small" higher fundamental forms as contained in scrolls.