abelian variety abstract algebraic variety affine space alexander grothendieck algebraic curve algebraic geometry algebraic manifold algebraic set algebraic space algebraic stack algebraically closed field birational geometry category theory closed set david cox david eisenbud dimension of an algebraic variety field of fractions function field fundamental theorem of algebra graph isomorphism homogeneous coordinates homogeneous polynomial identity map if and only if integral domain jean-pierre serre manifold mathematics nilpotent nilradical planetmath polynomial polynomial ring prime ideal projective algebraic manifold real number riemann sphere ring theory robin hartshorne sheaf cohomology singular point of an algebraic variety smooth function spectrum of a ring transcendence degree zariski topology