Local to Global Algorithms for the Gorenstein Adjoint Ideal of a Curve

Janko Böhm (), Wolfram Decker (), Santiago Laplagne () and Gerhard Pfister ()
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Janko Böhm: Technische Universität Kaiserslautern, Fachbereich Mathematik
Wolfram Decker: Technische Universität Kaiserslautern, Fachbereich Mathematik
Santiago Laplagne: Ciudad Universitaria, Departamento de Matemática, Facultad de Ciencias Exactas y Naturales
Gerhard Pfister: Technische Universität Kaiserslautern, Fachbereich Mathematik

A chapter in Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, 2017, pp 51-96 from Springer

Abstract: Abstract We present new algorithms for computing adjoint ideals of curves and thus, in the planar case, adjoint curves. With regard to terminology, we follow Gorenstein who states the adjoint condition in terms of conductors. Our main algorithm yields the Gorenstein adjoint ideal 𝔊 $$\mathfrak {G}$$ of a given curve as the intersection of what we call local Gorenstein adjoint ideals. Since the respective local computations do not depend on each other, our approach is inherently parallel. Over the rationals, further parallelization is achieved by a modular version of the algorithm which first computes a number of the characteristic p counterparts of 𝔊 $$\mathfrak {G}$$ and then lifts these to characteristic zero. As a key ingredient, we establish an efficient criterion to verify the correctness of the lift. Well-known applications are the computation of Riemann-Roch spaces, the construction of points in moduli spaces, and the parametrization of rational curves. We have implemented different variants of our algorithms together with Mnuk’s approach in the computer algebra system Singular and give timings to compare the performance.

Keywords: Adjoint ideals; Singularities; Algebraic curves; Primary 14Q05; Secondary 14H20, 14H50, 68W10 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/978-3-319-70566-8_3

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