Our research is motivated by recent work of Cook II, Harbourne, Migliore, and Nagel on configurations of points in the projective plane with properties that are unexpected from the point of view of the postulation theory. In this note, we revisit the basic configuration of nine points appearing in work of Di Gennaro/Ilardi/Vallès and Harbourne, and we exhibit some additional new properties of this configuration. We then pass to projective three-space P 3 and exhibit a surface with unexpected postulation properties there. Such higher dimensional phenomena have not been observed so far.If equality holds in (2), then we say that the system L N (d; m 1 , . . . , m s ) is non-special. Otherwise it is called special.

Abstract. The purpose of this note is to give defined over the real numbers counterexamples to a question relevant in the commutative algebra, concerning a containment relation between algebraic and symbolic powers of homogeneous ideal.

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The purpose of this note is to provide an overview of the containment problem for symbolic and ordinary powers of homogeneous ideals, related conjectures and examples. We focus here on ideals with zero dimensional support. This is an area of ongoing active research. We conclude the note with a list of potential promising paths of further research.