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]
The Rising Sea: Foundations of Algebraic Geometry by Ravi Vakil (2017 draft). Section 0.4, chapter 3 and rational-map development
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Ravi Vakil is a prominent mathematician known for his contributions to algebraic geometry, particularly in the areas of moduli spaces, deformation theory, and the theory of algebraic cycles. He is a professor at Stanford University and has published numerous influential papers and textbooks in the field. Vakil is also recognized for his work on the foundations of algebraic geometry, including his efforts to make complex topics more accessible to students through clear exposition and pedagogical methods.