The Intrinsic Hodge Theory of Hyperbolic Curves

In elementary mathematics, the simplest variety – i.e., geometric object defined by polynomial equations – that one encounters is a line, i.e., the set of points in the Euclidean plane R defined by an equation of the form aX + bY = c (where a, b, c ∈ R). After translation, rotation, and dilation, such an equation may be written in the form X = 0. In this case, the variety in question passes through the origin and, moreover, admits a natural group structure. Also, it is easy to understand in a very explicit way the totality of points (x, y) ∈ R that lie on this variety: Indeed, this set may be identified (via the projection (x, y) → y) with R itself. The next simplest type of variety that one encounters is the variety in R defined by an equation of degree 2. After translation, rotation, reflection, and dilation of the X and Y coordinates, we see that such an equation is always one of the following three types: X + Y 2 = 1, XY = 1, X = Y . In the final case, X = Y , the projection (x, y) → x defines a natural bijection of the set of points on the variety with R. In the first two cases, however, such projections do not give isomorphisms of the given variety with a linear variety.