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Unexpected curves, line arrangements, and Lefschetz properties Nagel, Uwe


We discuss connections between Lefschetz properties and the study of Hilbert functions of (fat) points as well as the theory of line arrangements. To this end, we begin by considering a finite set Z of points in the plane with the property that, for some integer j, the dimension of the linear system of plane curves of degree j + 1 through the points of Z and having multiplicity j at a general point is unexpectedly large. We give criteria for the occurrence of such unexpected curves and describe the range of their degrees. Inspired by work of Di Gennaro, Ilardi, and Vall`es, we relate properties of Z to properties of the arrangement of lines dual to the points of Z. In particular, we get a new interpretation of the splitting type of a line arrangement, and we show that the existence of an unexpected curve is equivalent to the failure of a certain Lefschetz property. This implies a Lefschetz-like criterion for Terao’s conjecture on the freeness of line arrangements. This is based on joint work with D. Cook II, B. Harbourne, and J. Migliore

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