Algebraic Geometry

This group gathers theorems from algebraic geometry, the study of the shapes defined by systems of polynomial equations. It covers Bezout's theorem on how many points two curves share, the Riemann-Roch and Riemann-Hurwitz theorems relating a curve's geometry to the functions that live on it, the Hodge index theorem constraining the intersection numbers on a surface, Chow's theorem linking analytic and algebraic varieties, and further results, including the Cayley-Bacharach theorem, that describe how the points where curves meet are themselves constrained by further algebraic conditions.