Algebraic geometry is one of the core areas of study in modern mathematics. It has a long history as the study of the geometric properties the zero sets of polynomials in multiple variables, and in its modern form brings in essentially all the tools provided by abstract algebra to better understand these zero sets and related objects. It has applications essentially everywhere these zero sets come up, and has applications in number theory, representation theory, differential and symplectic geometry, combinatorics, and in a variety of applied settings.
This course is an introduction to the ideas and objects of algebraic geometry. Topics include affine and projective varieties, dimension, smoothness, divisors, schemes, and the basic theory of projective curves. We will also build up a collection of example algebraic varieties that will come up again and again as you study more: Grassmannians, rational and elliptic curves, blowups, and low-degree hypersurfaces.
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