Curvas algebraicas y la pregunta de Halphen

TL;DR

This paper provides an elementary overview of the historical and mathematical context of Halphen's question regarding the genus and degree of algebraic curves in projective 3-space, including key contributions and recent developments.

Contribution

It offers a brief, accessible summary of the problem, its historical solutions, and modern research directions in the study of algebraic curves in projective space.

Findings

Historical solutions by Halphen, Castelnuovo, Gruson, and Peskin

Basic definitions of algebraic curves, genus, and degree

Discussion of modern developments and future research directions

Abstract

This is an expository paper written in Spanish. This paper discusses the answer and ideas around the Halphen question: what are the pairs (g,d) that occur as the genus and degree of a smooth algebraic curve in projective 3-space. Halphen's question has been studied by many authors, including Halphen in 1881, Castelnuovo in 1890, and was finally answered almost 100 years later by Gruson and Peskin in 1981. We (very briefly) sketch arguments by Castelnuovo in studying this question and comment on the ideas of the final answer provided by Gruson and Peskin. This paper is meant to be brief and elementary introduction to some ideas about algebraic curves embedded in projective space and as such, we start out with basic definitions such as that of an algebraic curve, genus and degree. Towards the end, we mention some modern developments about this question as well as directions of research.