Separators of points on algebraic surfaces

For a finite set of points image and for a given point image, the notion of a separator of P in image (a hypersurface containing all the points in image except P) and of the degree of P in X, image (the minimum degree of these separators) has been largely studied. In this paper we extend these notions to a set of points image on a projectively normal surface image, considering as separators arithmetically Cohen–Macaulay curves and generalizing the case image in a natural way. We denote the minimum degree of such curves as image and we study its relation to image. We prove that if S 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.