Space curves on surfaces with ordinary singularities

We show that smooth curves in the same biliaison class on a hypersurface P3 with ordinary singularities are linearly equivalent. compute invariants h0(IC(d)), h1(IC(d)), and h1(OC(d)) of curve C such surface X terms cohomologies divisors normalization X. then study general projections lying rational normal scroll S(a,b)⊂Pa+b+1. If we vary linear system S(a,b) as well projections, obtain family P3. dimension space deformations these family. difference is function b which does not depend system. Finally, classify maximal rank ruled cubic surfaces prove all but finitely many classes projectively S(1,2)⊂P4 fail to have These give infinitely counter-examples question Hartshorne.