Applications of iterated curve blowup to set-theoretic complete intersections in P3

Abstract

Let S, T be surfaces in P3. Suppose that S intersect T is set-theoretically a smooth curve C of degree d and genus g. Suppose that S and T have no common singular points. Then if C is not a complete intersection, then deg(S), deg(T) < 2d^4. Fixing (d,g), one can form a finite (shorter) list of all possible pairs (deg(S),deg(T)). For instance, when (d,g) = (4,0), and assuming for simplicity that deg(S) <= deg(T): (deg(S), deg(T)) \in {(3,4), (3,8), (4,4), (4,7), (6,26), (9,48), (10,28) (12,18), (13,16), (17,220), (18,118), (19,84), (20,67), (22,50), (28,33)}. Assume characteristic 0. [1] Suppose that S and T have non-overlapping rational singularities. Then d <= g+3. [2] Suppose that S is normal, and that d>deg(S). Then C is linearly normal (and so d <= g+3). [3] Suppose that S is a quartic surface having only rational singularities. Then C is linearly normal. Hard copy is available from the author. E-mail to [e-mail].