Local to global algorithms for the Gorenstein adjoint ideal of a curve

TL;DR

This paper introduces parallel algorithms for computing the Gorenstein adjoint ideal of algebraic curves, utilizing local-to-global strategies and modular methods to improve efficiency and enable applications in algebraic geometry.

Contribution

The paper presents novel parallel algorithms for Gorenstein adjoint ideals, including a modular approach with lift verification, advancing computational methods in algebraic geometry.

Findings

Algorithms are implemented in Singular and tested on various curves.

Parallelization significantly reduces computation time.

Modular approach with lift verification ensures correctness in characteristic zero.

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 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 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 of the algorithms.