The computation of the dimension of linear systems of plane curves through a bunch of given multiple points is one of the most classic issues in algebraic geometry. In general, it is still an open problem to understand when the points fail to impose independent conditions. Despite many partial results, a complete solution is not known, even if the fixed points are in general position. The answer in the case of general points in the projective plane is predicted by the famous Segre-Harbourne-Gimigliano-Hirschowitz conjecture. When we consider fixed points in special position, even more interesting situations may occur. Recently, Di Gennaro, Ilardi and Vallès discovered a special configuration Z of nine points with a remarkable property: A general triple point always fails to impose independent conditions on the ideal of Z in degree four. The peculiar structure and properties of this kind of unexpected curves were studied by Cook II, Harbourne, Migliore and Nagel. By using both explicit geometric constructions and more abstract algebraic arguments, we classify low-degree unexpected curves. In particular, we prove that the aforementioned configuration Z is the unique one giving rise to an unexpected quartic.

Starting with the pioneering work of Ein and Lazarsfeld [EinLaz93] restrictions on values of Seshadri constants on algebraic surfaces have been studied by many authors [Bau99, BauSze11, HarRoe08, KSS09, Nak05, Ste98, Sze12, Xu95].In the present note we show how approximation involving continued fractions combined with recent results of Küronya and Lozovanu on Okounkov bodies of line bundles on surfaces [KurLoz14, KurLoz15] lead to effective statements considerably restricting possible values of Seshadri constants. These results in turn provide strong additional evidence to a conjecture governing the Seshadri constants on algebraic surfaces with Picard number 1.

The purpose of this note is to study configurations of lines in projective planes over arbitrary fields having the maximal number of intersection points where three lines meet. We give precise conditions on ground fields F over which such extremal configurations exist. We show that there does not exist a field admitting a configuration of 11 lines with 17 triple points, even though such a configuration is allowed combinatorially. Finally, we present an infinite series of configurations which have a high number of triple intersection points.

In the present note we give a new proof of a result due to Wiseman and Wilson [13] which establishes an analogue of the Sylvester-Gallai theorem valid for curves of degree two. The main ingredients of the proof come from algebraic geometry. Specifically, we use Cremona transformation of the projective plane and Hirzebruch inequality (1).

Abstract.We study the Seshadri constants of ample line bundles on hyperelliptic surfaces. We obtain new lower bounds and compute the exact values of the Seshadri constants in some cases. Our approach uses results