"Riemann–Roch spaces and algebraic geometry codes" - Elena Berardini

Описание: "Riemann–Roch spaces and algebraic geometry codes" - Elena Berardini

Seminario del convegno UMI - DeCifris 2021. Il seminario è introdotto dalla professoressa Maria Tota.

Abstract: The Riemann–Roch space associated to a divisor on an algebraic variety is a finite dimensional vector space of rational functions which satisfy certain conditions on the location and order of their zeros and poles. Riemann–Roch spaces intervene in the construction of some error–correcting codes, called algebraic geometry codes, introduced by Goppa in 1981 on algebraic curves. Goppa’s construction naturally extends to algebraic varieties of any dimension and some work has been undertaken on surfaces.
In this talk we will approach Riemann–Roch spaces from two different points of view: the effective computation of a basis of these spaces in the case of curves, and their use in the study of algebraic geometry codes from surfaces. Recently, we proposed an algorithm for the computation of Riemann–Roch spaces of curves with any type of singularities. We will present the ideas behind this computation, based on Brill and Noether’s theory and Puiseux series expansions. The curves used in the construction of algebraic geometry codes are for the most part limited to those for which the Riemann–Roch spaces are already known. This new work will allow the construction of algebraic geometry codes from more general curves and divisors. This is far from being feasible in the context of surfaces. However, using some tools from algebraic geometry it is possible to study the Riemann–Roch space of a divisor on a surface and the parameters of the codes built from it. We will present these tools and give lower bounds for the dimension and the minimum distance of algebraic geometry codes from surfaces. This will allow us to discuss the interest of certain families of surfaces compared to others, in the context of algebraic geometry codes.
The results on non-ordinary curves are part of a work with S. Abelard, A. Couvreur and G. Lecerf. The results on codes from surfaces are part of two works with Y. Aubry, F. Herbaut and M. Perret.

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