A Theory of Divisors for Algebraic Curves

Details A Theory of Divisors for Algebraic Curves A Theory of Divisors for Algebraic Curves

2007-03-10

arxiv.org/abs/math/0703290v1

Mathematics

Algebraic Geometry

Scientific paper

The purpose of this paper is two-fold. We first prove a series of results, concerned with the notion of Zariski multiplicity, mainly for non-singular algebraic curves. These results are required in the paper "A Theory of Branches for Algebraic Curves",(*), where, following Severi, we introduced the notion of the "branch" of an algebraic curve. Secondly, we use results from the cited paper, (*), in order to develop a refined theory of g_{n}^{r} on an algebraic curve. The refinement depends critically on relacing the notion of a point with a branch. This allows us to construct a theory of divisors, \emph{generalising} the corresponding theory in the special case when the algebraic curve is non-singular, which is birationally invariant.

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