Roughly speaking, algebraic geometry studies the solution sets of systems of polynomial equations. One of the advantages of algebraic geometry is that it is purely algebraically defined and applied to any field, including fields of finite characteristic. It is geometry based on algebra rather than calculus, but over the real or complex numbers it provides a rich source of examples and inspiration to other areas of geometry.
- Affine algebraic varieties, coordinate rings, Hilbert's Nullstellensatz, the Zariski topology, morphisms of affine varieties, irreducible varieties, smooth and singular points.
- Projective space, projective varieties, the Zariski topology on projective varieties, morphisms of projective varieties, rational varieties and rational maps, linear systems and blow ups.
- Dimension of affine and projective varieties, Zariski tangent space.
Expected knowledge: Linear Algebra and Algebra 1 (rings and ideals).