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Algebraic geometry studies geometric objects defined by polynomial equations and the algebra of functions on them. For example, $y=x^2$ defines a parabola, while $xy=0$ defines the union of two coordinate lines. The coordinate ring $k[x,y]/(xy)$ records the second example algebraically: the images of $x$ and $y$ are nonzero but their product is zero, reflecting two components. Modern schemes extend this dictionary beyond ordinary point sets and permit geometric reasoning over general commutative rings. Applications range from elliptic curves in number theory to families of solutions depending on parameters.^[ref]

Ravi Vakil works on algebraic geometry. His Stanford page identifies the specialization and his exposition The Rising Sea: Foundations of Algebraic Geometry by Ravi Vakil (2017 draft).^[ref]

The book builds the geometry of polynomial equations through schemes, beginning with affine constructions and developing local-to-global methods. A rational map may be defined only on a dense open set; a birational morphism can change a geometric model while preserving its function field. These distinctions connect concrete equations with the abstract language needed to compare algebraic spaces.^[ref]