Space curves on surfaces with ordinary singularities

TL;DR

This paper investigates the properties of space curves on surfaces with ordinary singularities, computes their invariants, studies their projections from rational normal scrolls, and classifies maximal rank curves on ruled cubic surfaces, providing counterexamples to Hartshorne's question.

Contribution

It establishes linear equivalence of curves in the same biliaison class on such surfaces, computes key cohomological invariants, analyzes deformations under projections, and classifies maximal rank curves, including counterexamples.

Findings

Smooth curves in the same biliaison class are linearly equivalent.

Computed cohomological invariants of curves on surfaces with ordinary singularities.

Identified infinitely many classes of curves failing maximal rank in projections, countering Hartshorne's question.

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 $h^0(\mathscr{I}_C(d))$, $h^1(\mathscr{I}_C(d))$ and $h^1(\mathscr{O}_C(d))$ 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.