Abstract. The main goal of this paper is to show that the notions of Weil and Cartier Q-divisors coincide for V -varieties and give a procedure to express a rational Weil Divisor as a rational Cartier divisor. The theory is illustrated with weighted projective spaces and weighted blow-ups.
In this paper we study the intersection theory on surfaces with abelian quotient singularities and we derive properties of quotients of weighted projective planes. We also use this theory to study weighted blow-ups in order to construct embedded Q-resolutions of plane curve singularities and abstract Q-resolutions of surfaces.
In this paper we extend the concept of Milnor fiber and Milnor number to curve germs on surface quotient singularities. A generalization of the local δ-invariant is defined and described in terms of a Q-resolution of the curve singularity. In particular, when applied to the classical case (the ambient space is a smooth surface) one obtains a formula for the classical δ-invariant in terms of a Q-resolution, which simplifies considerably effective computations. All these tools will finally allow for an explicit description of the genus formula of a curve defined on a weighted projective plane in terms of its degree and the local type of its singularities.
We obtain an approximate solutionẼ =Ẽ(e, M ) of Kepler's equation E − e sin(E) = M for any e ∈ [0, 1) and M ∈ [0, π]. Our solution is guaranteed, via Smale's α-theory, to converge to the actual solution E through Newton's method at quadratic speed, i.e. the n-th iteration produces a value En such that |En −E| ≤ ( 1 2 ) 2 n −1 |Ẽ −E|. The formula provided forẼ is a piecewise rational function with conditions defined by polynomial inequalities, except for a small region near e = 1 and M = 0, where a single cubic root is used. We also show that the root operation is unavoidable, by proving that no approximate solution can be computed in the entire region [0, 1) × [0, π] if only rational functions are allowed in each branch. √ 1 − e 2 and T can be obtained from a using the third law). For a derivation of these formulas, and a detailed introduction to Kepler's equation, see [1].By a symmetry argument, the equation can be easily reduced to the case M ∈ [0, π]. The existence and uniqueness of solution E ∈ [0, π] follows from the fact that the function f e,M : [0, π] → [0, π] given by f e,M (E) = E − e sin(E) − M is strictly increasing.Several solutions to the problem have been proposed since it was stated 400 years ago. Some authors have tried non-iterative methods to solve the equation up to a fixed predetermined accuracy ([6]; [8]). However, we want to calculate the solution with arbitrary precision, hence our interest in iterative techniques.Kepler himself proposed to use a fixed-point iteration to solve the equation ([3, Ch. 1]), i.e. guess E 0 , an approximation of the exact solution E, and then iterate E n+1 = M + e sin(E n ). This sequence converges to E, sinceThe problem with this approach is that the convergence is slow for values of e near 1. For the orbit of Mercury, which has e ≈ 0.2, about 5 iterations are needed to reduce the error by a factor of 10 −3 , while for values of eccentricity e > 0.5 the fixed-point iteration is even slower than a bisection method.Although the fixed-point iteration does not provide an efficient solution to Kepler's equation, it exhibits the structure of most of the current methods to solve it: first, guess an approximationẼ of the solution (called starter ), and then use some iterative technique to produce a sequence quickly converging to the actual solution (see [4], [5], [9], [13]). For the second part, Newton's method seems to be the most used iteration, mainly due to its conceptual simplicity, generality and fast convergence. The guessing part, however, requires some specific understanding on the equation and has been the subject of many recent papers ([2]; [7]; [10]; [11]; [12]; [15]).Starters have been compared (and optimized) using different criteria, such as the number of iterations needed to reach certain precision, the distance to the actual solution, the number of floating point operations needed for its computation, etc. For this purpose, we adopt a criterion which is very specific to Newton's method and guarantees that the iterations reduce the error at quadratic sp...
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