A Theory of Branches for Algebraic Curves (original) (raw)

Zariski Structures and Algebraic Geometry

2004

The purpose of this paper is to provide a new account of multiplicity for finite morphisms between smooth projective varieties. Traditionally, this has been defined using commutative algebra in terms of the length of integral ring extensions. In model theory, a different approch to multiplicity was developed by Zilber using the techniques of non-standard analysis. Here, we first reformulate Zilber's method in the language of algebraic geometry and secondly show that, in classical projective situations, the two notions essentially coincide. As a consequence, we can recover intersection theory in all characteristics from the non-standard method and sketch further developments in connection with etale cohomology and deformation theory. The usefulness of this approach can be seen from the increasing interplay between Zariski structures and objects of non-commutative geometry.

A Theory of Divisors for Algebraic Curves

2007

The purpose of this paper is twofold. 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 [6], where, following Severi, we introduced the notion of the "branch" of an algebraic curve. Secondly, we use results from [6], in order to develop a refined theory of g r n on an algebraic curve. This refinement depends critically on replacing the notion of a point with that of a "branch". We are then able to construct a theory of divisors, generalising the corresponding theory in the special case when the algebraic curve is non-singular, which is birationally invariant.

Algebraic geometry (Selected Topics)

This topic contains lecture notes from one-semester course for graduate students which took place in Mathematical institute SANU in the spring semester 1994/95. It includes basic notions of algebraic geometry such as plane algebraic curves, Weil and Cartier divisors, dimension, sheaves and Czech cohomology, topological and arithmetical genus, linear systems, Riemann-Roch theorem for curves.

Algebraic Geometry – An Introduction . Daniel Perrin

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A Theory of Duality for Algebraic Curves

Eprint Arxiv 1003 5837, 2010

We formulate a refined theory of linear systems, using the methods of a previous paper, "A Theory of Branches for Algebraic Curves", and use it to give a geometric interpretation of the genus of an algebraic curve. Using principles of duality, we prove generalisations of Plucker's formulae for algebraic curves. The results hold for arbitrary characteristic of the base field L, with some occasional exceptions when characteristic(L)=2, which we observe in the course of the paper.

On the branches of a complex space

Journal für die reine und angewandte Mathematik (Crelles Journal), 1981

at Bologna Introduction. In this note we give a geometric Interpretation for the number of geometric branches (defmed by S. Greco in [4]) of the local ring of a point of a complex space, localized at a prime ideal. Precisely (cf. prop. l and thm. 2), if & x x is such a local ring, p one of its prime ideals, Υ an analytic subset in an open neighbourhood of x, which induces the germ associated to p, then the number of the geometric branches of (d) x x \ counts the analytic branches of X at each point of an open dense subset of a suitable compact neighbourhood of χ in Y. The proof of this result (which answers a question posed by J. Lipman) is based on the properties of the ring of holomorphic functions on a compact semianalytic Stein subset of a complex analytic space, studied by J. Frisch in [3] and Greco-Traverso in [5]. Later on we globalize the previous result showing (cf. prop. 6) that if X is a reduced complex space, and Υ an irreducible analytic subset of X, there exists a Zariski open dense subset W<^Y such that the number of analytic branches of X at each point of W is constant (and we shall denote its value by g(Y)). Moreover (cf. thm. 7), for all xe Y, denoting by p^ the prime ideal of & x x associated to the germ Y x , the number of the geometric branches of the ring (& XfX) px is constant and its value is g(Y). This result is the analogue for analytic spaces of a result of S. Greco for algebraic varieties (cf. [4], (1.6)). As an application of theorem 7 we prove (cf. prop. 8) the analogue for complex spaces of a result by D. Ferrand for schemes (cf. [1], (4. 10)), which also gives a criterion for a complex space to be S 2. The authors wish to thank Professor O. Forster for his helpful suggestions. Foreword. In the following "complex space" and "analytic subset" always mean "complex analytic space" and "closed analytic subset" respectively. Preliminaries. Recall that if A" is a reduced scheme, and/: X-* X its normalization, a geometric branch of X at its point x is a geometric point of the fc(x)-scheme f" 1 (x) (cf. [4], par. 1).