Many algebraic varieties are manifolds, but
an algebraic variety may have singular points while a manifold cannot.
Algebraic varieties can be characterized by their dimension.
Algebraic varieties of dimension one are called algebraic curves and
algebraic varieties of dimension two are called algebraic surfaces.
In the context of modern scheme theory,
an algebraic variety over a field is an integral (irreducible and reduced) scheme over that field
whose structure morphism is separated and of finite type.
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