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In a paper by J. V. Pereira and the author it was proved that any pencil of plane curves of degree $d>1$ with irreducible generic fiber can have at most five completely reducible fibers although no examples with five such fibers had ever been found. Recently Janis Stipins has proved that if a pencil has a base of $d^2$ points, then it cannot have five completely reducible fibers. In this paper we generalize Stipins' result to arbitrary pencils. We also include into consideration more general special fibers that are the unions of lines and non-reduced curves. These fibers are important for characteristic varieties of hyperplane complements. [ABSTRACT FROM AUTHOR]

Abstract: For a finite set of points and for a given point , the notion of a separator of in (a hypersurface containing all the points in except ) and of the degree of in , (the minimum degree of these separators) has been largely studied. In this paper we extend these notions to a set of points on a projectively normal surface , considering as separators arithmetically Cohen–Macaulay curves and generalizing the case in a natural way. We denote the minimum degree of such curves as and we study its relation to . We prove that if is a variety of minimal degree these two terms are explicitly related by a formula, whereas only an inequality holds for other kinds of surfaces. [Copyright &y& Elsevier]