One of the main themes of "modern" algebraic geometry is the study of $families$ of algebraic varieties; in fact, just consider the huge subjects known as deformation theory and moduli spaces theory. This leads very naturally (at least from our "modern" point of view) to situations where the knowledge of schemes (non-reduced structures), commutative algebra (flatness, Cohen-Macauleyness, etc) and homological algebra are essential not only from a theoretical point of view, but also in order to make explicit computations with very concrete objects, e.g. quasi-projective varieties. Such "modern" tools have undoubtedly made the study of algebraic geometry harder for the beginner, but on the other hand they have brought clarity in many situations where the "classical methods" did not work well.

One of the main themes of "modern" algebraic geometry is the study of families of algebraic varieties; in fact, just consider the huge subjects known as deformation theory and moduli spaces theory. This leads very naturally (at least from our "modern" point of view) to situations where the knowledge of schemes (non-reduced structures), commutative algebra (flatness, Cohen-Macauleyness, etc) and homological algebra are essential not only from a theoretical point of view, but also in order to make explicit computations with very concrete objects, e.g. quasi-projective varieties. Such "modern" tools have undoubtedly made the study of algebraic geometry harder for the beginner, but on the other hand they have brought clarity in many situations where the "classical methods" did not work well.