zbMATH — the first resource for mathematics

The genus of space curves. (English) Zbl 0865.14013
The maximal value for the arithmetic genus of projective curves of degree \(d\) is classically known to be \((d-1)(d-2)/2\), i.e. the genus of smooth plane curves; this remains true even if the word “curve” means any locally Cohen-Macaulay scheme of dimension 1. For irreducible, non degenerate curves \(C\) in \(\mathbb{P}^r\), Castelnuovo obtained a sharp bound for the arithmetic genus; however, when we consider also reducible curves, possibly with isolated points, Castelnuovo’s bound is no longer valid, and the question about bounds for the genus of these objects is more complicate. For arbitrary (non degenerate) locally Cohen Macaulay curves in \(\mathbb{P}^3\), Okonek and Spindler used the spectrum of associated torsion free sheaves, to prove that the genus is bounded by \((d-2)(d-3)/2\). The author presents here a new proof of this result, which uses the classical Castelnuovo technique of reduction to general plane sections; he also provides examples of non degenerate curves of degree \(d\) and genus \(g\) for any pair \((d,g)\) satisfying \(g\leq(d-2) (d-3)/2\), and classifies extremal examples.

14H45 Special algebraic curves and curves of low genus
14H50 Plane and space curves