Space curves on surfaces with ordinary singularities

Abstract

We show that smooth curves in the same biliaison class on a hypersurface in ${\mathbf{P}}^3$ with ordinary singularities are linearly equivalent. We compute the invariants $h^0({\mathcal{I}}_C(d))$, $h^1({\mathcal{I}}_C(d))$, and $h^1({\mathcal{O}}_C(d))$ 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 ${\mathbf{P}}^3$ of curves lying on the rational normal scroll $S(a,b)\subset {\mathbf{P}}^{a+b+1}$. If we vary the curves in a linear system on $S(a,b)$ as well as the projections, we obtain a family of curves in ${\mathbf{P}}^3$. We compute the dimension of the space of deformations of these curves in ${\mathbf{P}}^3$ 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 ${\mathbf{P}}^3$. We prove that the general projections of all but finitely many classes of projectively normal curves on $S(1,2)\subset {\mathbf{P}}^4$ fail to have maximal rank in ${\mathbf{P}}^3$. These give infinitely many classes of counter-examples to a question of Hartshorne.