Gorenstein Liaison of curves in P

Let k be an algebraically closed field of characteristic zero, S = k[X0, X1, X2, X3, X4] and P = Proj(S). By a curve we always mean a closed one-dimensional subscheme of P which is locally Cohen-Macaulay and equidimensional. The main purpose of this paper is to show that arithmetically Cohen-Macaulay curves C ⊂ P lying on a “general” arithmetically Cohen-Macaulay surface X ⊂ P with degree matrix [ui,j], ui,j > 0, are glicci provided 16((KH) 2 − KH) − H[H − K + 8(1 + pa)] ≥ 0; being K the canonical divisor on X and H the hyperplane section of X. We also give examples of arithmetically Cohen-Macaulay surfacesX ⊂ P verifying the above numerical condition. The idea of using complete intersections to link varieties has been used for a long time, going back at least to work of Macaulay and Severi. Since them, many mathematicians have contributed to the development of liaison theory and we have a remarkable picture of the liaison theory and its applications in codimension 2. One naturally would like to carry out a program in higher codimension. In [KMMNP], the authors demonstrate that Gorenstein liaison is a natural generalization of the wellunderstood theory of CI-liaison codimension two cases. In particular, in [KMMNP]; Theorem 3.6 it is proved that every standard determinantal scheme X ⊂ P is glicci (i.e. X belongs to the Gorenstein liaison class of a complete intersection), and it is posed the following question: