Ruled and rational surfaces and their models

TL;DR

This paper surveys the concept of models in algebraic geometry, focusing on rational and ruled surfaces over complex fields, to facilitate their classification and study.

Contribution

It defines and analyzes models of rational and ruled surfaces, providing a framework for understanding their structure and properties in the smooth complex setting.

Findings

Develops a comprehensive definition of models for these surfaces

Provides methods to classify and study ruled and rational surfaces using models

Enhances understanding of surface properties through model analysis

Abstract

One of the most powerful ideas in the study and classification of algebraic varieties is the notion of a model: that is, to single out an object, in the appropriate isomorphism class, with nice properties. This survey aims to define and study suitable models of rational and, more generally, ruled surfaces in the smooth complex case, and to use them to study such surfaces.