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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]

By Ravi Vakil. The title page identifies the November 18, 2017 draft. Author-hosted PDF: math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf

Consulted passages: §0.4 (goals and polynomial equations), Chapter 3 (affine schemes), and the sections on rational maps and birational morphisms.