Let S be an algebraic surface that is non-singular and projective.

Not every singular algebraic surface is a scheme.

He worked on algebraic surfaces and later in his career became interested in elliptic functions.

For more examples see the list of algebraic surfaces.

It is a compact smooth algebraic surface, but is not in general minimal.

The papers , and identified their type in the classification of algebraic surfaces.

Such objects are always algebraic surfaces of general type.

However they are not algebraic surfaces (except in special cases).

This notion works perfectly for algebraic surfaces (varieties of dimension 2).

That theory is too limited for algebraic surfaces, and even for curves with singular points.