Curves on Normal Surfaces

We study locally Cohen-Macaulay space curves lying on normal surfaces. We prove some theorems on the behaviour of the cohomology functions and initial ideals of such space curves, which give a basic distinction between those curves and curves lying on non-normal surfaces. A natural step in the study and classification of space curves is to classify the Hilbert functions of space curves, or, more generally, cohomology functions of space curves. A closely related approach is to study the somewhat more refined invariants, the generic initial ideal and higher initial ideal, and classify which of them can occur for smooth, integral or locally Cohen-Macaulay space curves. Given a space curve C on a surface S ⊆ P, one may do descending biliaison on the surface S and end up with a curve X minimal in its biliaison class on the surface S. On the other hand, one might consider a surface T of minimal degree such that C may be linked (S, T ) to a curve Y . In these cases it will often happen thatX or Y will have non-reduced components. The invariants of C may be deduced from the invariants of X or Y . Hence it is natural to try to get an understanding of the invariants of curves on S with nonreduced components. Here it turns out that there is a basic distinction between when S is normal and when S is not normal. The reason is that if one has a reduced curve D on S then a multiplicity n structure on D in S is uniquely determined if S is normal, while if S is not normal there may exist many (in fact infinitely many) multiplicity n structures on D in S. For C a curve on a surface S of degree s there is a basic sequence 0 −→ L −→ OS −→ OC −→ 0, (1) where L is the ideal sheaf of C in S. This also gives rise to a sequence 0 −→ OS(s− 4) −→ L∨(s− 4) −→ ωC −→ 0. (2) In both cases L and L∨(s − 4) are torsion free rank one sheaves on S with local projective dimension one as OP3-modules. We prove the following. Theorem 2.1. Let S ⊆ P be a normal surface of degree s, and L a torsion free rank one sheaf on S with local projective dimension one as OP3-module. Let the minimal generators of H ∗L= ⊕ n∈ZH L(n) occur in degrees n0 ≤ n1 ≤ · · · ≤ nr. Then nj − nj−1 ≤ s− 2 for j = 1, . . . , r, and the regularity of L is ≤ nr + (s− 2). Received by the editors July 4, 1996 and, in revised form, September 22, 1997. 2000 Mathematics Subject Classification. Primary 14H50.