Linear systems of rational curves on rational surfaces

TL;DR

This paper characterizes the set of linear systems on rational surfaces that contain a given rational curve and whose general members are also rational, providing a complete description of such systems.

Contribution

It offers a comprehensive classification of linear systems containing a specific rational curve on rational surfaces, identifying when such systems are non-empty.

Findings

Characterization of linear systems containing a given rational curve

Criteria for the non-emptiness of these linear systems

Complete description of the set Omega_C

Abstract

Given a curve C on a projective nonsingular rational surface S, over an algebraically closed field of characteristic zero, we are interested in the set Omega_C of linear systems Lambda on S satisfying C is in Lambda, dim Lambda > 0, and the general member of Lambda is a rational curve. The main result of the paper gives a complete description of Omega_C and, in particular, characterizes the curves C for which Omega_C is non empty.