1. Research on algebraic curves and surfaces: This is the most basic research object in algebraic geometry, including plane curves and spatial curves and surfaces.
2. Research on algebraic cluster: Algebraic cluster is a set defined by a set of polynomial equations, and it is one of the most important research objects in algebraic geometry.
3. Research on algebraic topology: Algebraic topology is an important field where algebraic geometry and topology intersect, and mainly studies the topological properties of algebraic clusters.
4. Research on arithmetic geometry: Arithmetic geometry is an important field where algebraic geometry and number theory intersect, and it mainly studies the properties of algebraic clusters over integer rings.
5. Application of Algebraic Geometry: Algebraic Geometry has applications in many fields, such as physics, computer science, engineering and so on.
6. Other research directions: In addition to the above directions, algebraic geometry has many other research directions, such as birational geometry, nonsingular algebraic geometry, projective algebraic geometry and so on.