Birational classification of curves on rational surfaces

TL;DR

This paper classifies pairs of rational surfaces and linear systems, providing a theorem for their birational classification and identifying minimal models of plane curves under Cremona transformations.

Contribution

It introduces a classification theorem for pairs (S, L) and determines Cremona minimal models for irreducible plane curves, advancing understanding of birational geometry.

Findings

Classification theorem for pairs (S, L) on rational surfaces

Determination of Cremona minimal models for plane curves

Explicit methods for minimal degree models under Cremona transformations

Abstract

In this paper we consider the birational classification of pairs (S,L), with S a rational surfaces and L a linear system on S. We give a classification theorem for such pairs and we determine, for each irreducible plane curve B, its "Cremona minimal" models, i.e. those plane curves which are equivalent to B via a Cremona transformation, and have minimal degree under this condition.