We show that smooth curves in the same biliaison class on a hypersurface in with ordinary singularities are linearly equivalent. We compute the invariants , , and of a curve C on such a surface X in terms of the cohomologies of divisors on the normalization of X. We then study general projections in of curves lying on the rational normal scroll . If we vary the curves in a linear system on as well as the projections, we obtain a family of curves in . We compute the dimension of the space of deformations of these curves in as well as the dimension of the family. We show that the difference is a linear function in a and b which does not depend on the linear system. Finally, we classify maximal rank curves on ruled cubic surfaces in . We prove that the general projections of all but finitely many classes of projectively normal curves on fail to have maximal rank in . These give infinitely many classes of counter-examples to a question of Hartshorne.
"Space curves on surfaces with ordinary singularities." Illinois J. Math. 66 (1) 1 - 29, April 2022. https://doi.org/10.1215/00192082-9753971