On Hodge Theory of Singular Plane Curves

On Hodge Theory of Singular Plane Curves

The dimensions of the graded quotients of the cohomology of a plane curve complement U = P \ C with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail. We also give a precise numerical estimate for the difference between the Hodge filtration and the pole order filtration on H(U,C).

Singular Points of Plane Curves

ly isomorphic to (C×)r−1 × (C), and hence also to (S1)r−1 × (R), where r = |J | is the number of branches and k = δ(C)− r+1 = 1 2 (μ(C) + 1 − r). The construction of the Jacobian variety J(C̃) of the non-singular curve C̃ in the large is standard in algebraic geometry. There is also a notion of Jacobian of a singular curve C , defined e.g. in [85], which, like the other, is an abelian group. Ther...

The Hodge - Arakelov Theory of Elliptic Curves

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 thr...

Polars of Real Singular Plane Curves

Polar varieties have in recent years been used by Bank, Giusti, Heintz, Mbakop, and Pardo, and by Safey El Din and Schost, to find efficient procedures for determining points on all real components of a given non-singular algebraic variety. In this note we review the classical notion of polars and polar varieties, as well as the construction of what we here call reciprocal polar varieties. In p...