Curves on normal rational cubic surfaces

John Brevik

Vol. 230 (2007), No. 1, 73–105

DOI: 10.2140/pjm.2007.230.73

Abstract

Let k be an algebraically closed field, let X₀ be a rational normal cubic surface in ℙ³ = ℙ_k³, and let C₀ ⊂ X₀ be a locally Cohen–Macaulay curve, which is therefore an effective Weil divisor on X₀. I show that C₀ can be expressed as the limit of a family of curves whose general member lies on a smooth surface, in the following sense: There exists a flat family X_t of cubic surfaces specializing to X₀ and a flat family C_t of curves specializing to C₀, parametrized by a smooth (noncomplete) curve T, such that the general member of X_t is a smooth cubic surface and C_t ⊂ X_t is an effective (Cartier) divisor for all t ∈ T ∖{0}.

Keywords

classification of space curves, curves on singular surfaces, singular del Pezzo surfaces

Mathematical Subject Classification 2000

Primary: 14H50

Secondary: 14J17

Milestones

Received: 19 August 2005

Revised: 9 January 2006

Published: 1 March 2007