Calling $X$ a variety would appear to conflict with the preexisting notion of an affine or projective variety. However, it can be shown that if $k$ is algebraically closed, then there is an equivalence of categories between affine abstract varieties over $k$ and affine varieties over $k$ , and another between projective abstract varieties over $k$ and projective varieties over $k$ .

This equivalence of categories identifies an abstract variety with the set of its $k$ -points; this can be thought of as simply ignoring all the generic points. In the other direction, it identifies an affine variety with the prime spectrum of its coordinate ring: the variety in $\mathbb{A}^n$ defined by the ideal$$ \left<f_1,\ldots,f_m\right>$$ is identified with$$ \Spec k[X_1,\ldots,X_n]/\left<f_1,\ldots,f_m\right>.$$

A projective variety is identified as the gluing together of the affine varieties obtained by taking the complements of hyperplanes. To see this, suppose we have a projective variety in $\mathbb{P}^n$ given by the homogeneous ideal $\left<f_1,\ldots,f_m\right>$ . If we delete the hyperplane $X_i=0$ , then we obtain an affine variety: let $T_j = X_j/X_i$ ; then the affine variety is the set of common zeros of$$ \left<f_1(T_0,\ldots,T_n),\ldots,f_m(T_0,\ldots,T_n)\right>.$$ In this way, we can get $n+1$ overlapping affine varieties that cover our original projective variety. Using the theory of schemes, we can glue these affine varieties together to get a scheme; the result will be projective.

For more on this, see Hartshorne's book Algebraic Geometry; see the bibliography for algebraic geometry for more resources.