This subtopic studies foundations of algebraic geometry, including the basic language and principles that organize the subject across examples and general theory.

This subtopic studies local theory of algebraic varieties, emphasizing singularity, local structure, and the geometry of neighborhoods of points.

This subtopic studies cycles and subschemes, analyzing algebraic cycles, divisors, and the geometry of closed subschemes inside varieties.

This subtopic studies families and moduli, focusing on parameter spaces, deformations, and how geometric objects vary in families.

This subtopic studies birational geometry, where maps that are isomorphisms away from lower-dimensional subsets are used to compare varieties.

This subtopic studies cohomological methods in algebraic geometry, including sheaf cohomology, derived categories, and cohomological invariants.

This subtopic studies arithmetic problems in algebraic geometry, linking geometry with rational points, local-global principles, and number-theoretic questions.

This subtopic studies curves, covering their geometry, moduli, singularities, maps, and the rich interaction between curve theory and arithmetic.

This subtopic studies surfaces and higher-dimensional varieties, emphasizing classification, birational geometry, and the structure of geometric objects in higher dimension.

This subtopic studies abelian varieties and complex tori, examining their geometry, endomorphisms, and deep arithmetic and analytic structure.

This subtopic studies algebraic groups, including linear algebraic groups, group actions, and the interplay between geometry and group structure.

This subtopic studies special varieties, including rational, Fano, Calabi–Yau, and other distinguished classes that play central roles in modern geometry.

This subtopic studies projective and enumerative geometry, where incidence relations and counting problems are studied through geometric methods.

This subtopic studies real algebraic and analytic geometry, focusing on real varieties, semialgebraic sets, and geometric behavior over the reals.

This subtopic studies tropical geometry, where combinatorial shadow methods are used to study algebraic geometry through piecewise-linear structures.