Minimal families of curves on surfaces

TL;DR

This paper classifies minimal families of rational curves on surfaces, especially weak Del Pezzo surfaces, providing a systematic way to compute and understand such families and their applications to surface parametrizations.

Contribution

It introduces a constructive classification of minimal families of curves on surfaces, particularly on weak Del Pezzo surfaces, and applies this to surface parametrization problems.

Findings

Classified minimal families on weak Del Pezzo surfaces.

Provided a table of minimal families up to Weyl equivalence.

Generalized results on surfaces with families of conics.

Abstract

A minimal family of curves on an embedded surface is defined as a 1-dimensional family of rational curves of minimal degree, which cover the surface. We classify such minimal families using constructive methods. This allows us to compute the minimal families of a given surface. The classification of minimal families of curves can be reduced to the classification of minimal families which cover weak Del Pezzo surfaces. We classify the minimal families of weak Del Pezzo surfaces and present a table with the number of minimal families of each weak Del Pezzo surface up to Weyl equivalence. As an application of this classification we generalize some results of Schicho. We classify algebraic surfaces which carry a family of conics. We determine the minimal lexicographic degree for the parametrization of a surface which carries at least 2 minimal families.