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

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Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory

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 \(\mathfrak {G}\) 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 \(\mathfrak {G}\) 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.

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Notes

  1. 1.

    The term point will always refer to a closed point.

  2. 2.

    Recall that an ordinary multiple point of multiplicity i is a singularity where the lowest non-vanishing jet of f factors into i distinct linear factors.

  3. 3.

    The notation O(m) stands for terms of degree ≥ m.

  4. 4.

    In our context, a prime p is bad if Algorithm 3, applied to the modulo p values of the input over the rationals, does not return the reduction of the characteristic zero result.

  5. 5.

    We have to use a weighted cardinality count: when enlarging \(\mathscr {P}\), the total weight of the elements already present must be strictly smaller than the total weight of the new elements. Otherwise, though highly unlikely in practical terms, it may happen that only unlucky primes are accumulated.

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Acknowledgements

We would like to thank Gert-Martin Greuel, Christoph Lossen, Thomas Markwig, Mathias Schulze, and Frank Seelisch for helpful discussions.

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Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Kaiserslautern, Postfach 3049, 67653, Kaiserslautern, Germany

    Janko Böhm, Wolfram Decker & Gerhard Pfister

  2. Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria, 1428, Pabellón I, Buenos Aires, Argentina

    Santiago Laplagne

Authors
  1. Janko Böhm
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  2. Wolfram Decker
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  3. Santiago Laplagne
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  4. Gerhard Pfister
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Correspondence to Janko Böhm .

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Editors and Affiliations

  1. IWR, Heidelberg University, Heidelberg, Germany

    Gebhard Böckle

  2. Department of Mathematics, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Wolfram Decker

  3. Department of Mathematics, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Gunter Malle

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Böhm, J., Decker, W., Laplagne, S., Pfister, G. (2017). Local to Global Algorithms for the Gorenstein Adjoint Ideal of a Curve. In: Böckle, G., Decker, W., Malle, G. (eds) Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-70566-8_3

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