Separators of points on algebraic surfaces

For a finite set of points X⊆Pn and for a given point P∈X, the notion of a separator of P in X (a hypersurface containing all the points in X except P) and of the degree of P in X, dXP (the minimum degree of these separators) has been largely studied. In this paper we extend these notions to a set of points X on a projectively normal surface S⊆Pn, considering as separators arithmetically Cohen-Macaulay curves and generalizing the case S=P2 in a natural way. We denote the minimum degree of such curves as dX,SP and we study its relation to dXP. 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.