Curves of degree d < 6 in P³

Introduction

The reader will find a fairly complete description of curves of degree <6 in P³ (3-dimensional projective space over an algebraically closed field, k, with ch(k) = 0).

Particular attention is given to: postulation (cohomology of the ideal sheaf), minimal free resolution of the graded ideal, plane sections, multisecants.

The main goal here are not the results (which are well known), but to introduce some of the basic techniques in the study of space curves (or more generally, of projective varieties).

• Curves of degree 3 (the twisted cubic).

• Curves of degree 4.

Rational quartic curves.
Elliptic quartics curves.


• Curves of degree 5.

Rational quintic curves.
Elliptic quintic curves.
Curves of degree 5, genus 2.

 

Some general facts.

Generalities on minimal free resolutions.
The minimal free resolution of a curve on a quadric..
The lemma of Castelnuovo-Mumford.
Projectively normal curves.