Doing classifications is an old tradition in the field of Algebraic Geometry. It is a natural starting point when trying to understand the geometry of algebraic objects.

A classification consists of

• A set X of objects to study. This can be, for example, the set of cubic surfaces (surfaces of degree 3) in the projective space.
• An equivalence relation that tells us what objects are considered to be of the same type. For example, two surfaces can be considered equivalent if one is a projective transformation of the other.
• A description of all the object types. This can be a list if the number is finite, but the set of types often contains a parameterized collection of types.

Note that changing the set X or the equivalence relation may make it much harder or much easyer to compile a list of object types.

For example, note that the description of all affine cubic surfaces up to affine coordinate changes will be much longer than the description of all projective cubic surfaces up to projective equivalence. However, since any affine cubic surface can be found from a projective cubic surface, mathematicians has not found it interesting to produce the very long list of affine cubic surfaces. For information on cubic surfaces, see www.cubicsurface.net and the references therin, or the report from the GAIA project.

As another example, curves in the plane are much easier to classify when the ground field is the complex numbers C, compared to when it is the real numbers R.

In many cases we have a complete understanding of the projective complex case, and less understanding of the real affine case.

Work package 4.1 of the GAIA II project deals with classification. In most cases we have found it appropriate to study the real projective case, or, if this has prooven too hard, the complex projective case. This is partly because the number of cases explode when going from a projective to a affine study, and partly because it should be quite easy (but tiresome) to extend the study to the affine case.