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

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.

[1]  D. Le Brigand,et al.  Algorithme de Brill-Noether et codes de Goppa , 1988 .

[2]  Gerhard Pfister,et al.  Computing integral bases via localization and Hensel lifting , 2015, J. Symb. Comput..

[3]  D. Eisenbud.  Commutative Algebra: with a View Toward Algebraic Geometry , 1995 .

[4]  W. W. Adams,et al.  An Introduction to Gröbner Bases , 2012 .

[5]  K. Petri.  Über Spezialkurven. I. , 1925 .

[6]  L. O'carroll.  AN INTRODUCTION TO GRÖBNER BASES (Graduate Studies in Mathematics 3) , 1996 .

[7]  M'hammed El Kahoui,et al.  An Algorithm to Compute the Adjoint Ideal of an Affine Plane Algebraic Curve , 2014, Math. Comput. Sci..

[8]  S. Greco,et al.  On the theory of adjoints II , 1982 .

[9]  Irena Swanson,et al.  An algorithm for computing the integral closure , 2009, 0901.0871.

[10]  F. Orecchia,et al.  On the computation of the adjoint ideal of curves with ordinary singularities , 2014 .

[11]  Peter Kornerup,et al.  Mapping integers and hensel codes onto Farey fractions , 1982, BIT.

[12]  Andreas Steenpaß,et al.  Parallel algorithms for normalization , 2011, J. Symb. Comput..

[13]  John Milnor,et al.  Singular Points of Complex Hypersurfaces. (AM-61), Volume 61 , 1969 .

[14]  Heisuke Hironaka,et al.  On the arithmetic genera and the effective genera of algebraic curves , 1957 .

[15]  O. Keller.  Vorlesungen über algebraische Geometrie , 1974 .

[16]  P. Griffiths,et al.  Geometry of algebraic curves , 1985 .

[17]  G. Greuel,et al.  The Milnor number and deformations of complex curve singularities , 1980 .

[18]  Oscar Zariski,et al.  Commutative Algebra , 1958 .

[19]  Hans Schönemann,et al.  SINGULAR: a computer algebra system for polynomial computations , 2001, ACCA.

[20]  J. Rafael Sendra,et al.  Parametrization of Algebraic Curves over Optimal Field Extensions , 1997, J. Symb. Comput..

[21]  Craig Huneke,et al.  Integral closure of ideals, rings, and modules , 2006 .

[22]  B. Riemann,et al.  Theorie der Abel'schen Functionen. , 1857 .

[23]  Gerhard Pfister,et al.  The Normalization: a new Algorithm, Implementation and Comparisons , 1999 .

[24]  Elizabeth A. Arnold,et al.  Modular algorithms for computing Gröbner bases , 2003, J. Symb. Comput..

[25]  Mark van Hoeij,et al.  An Algorithm for Computing an Integral Basis in an Algebraic Function Field , 1994, J. Symb. Comput..

[26]  G. Greuel,et al.  A Singular Introduction to Commutative Algebra , 2002 .

[27]  Michal Mnuk.  An Algebraic Approach to Computing Adjoint Curves , 1997, J. Symb. Comput..

[28]  G. Castelnuovo.  Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica , 1893 .

[29]  Gerhard Pfister,et al.  Local Analytic Geometry , 2013 .

[30]  G. Castelnuovo.  Massima dimensione dei sistemi lineari di curve piane di dato genere , 1890 .

[31]  Francesco Severi,et al.  Vorlesungen über Algebraische Geometrie: Geometrie auf einer Kurve Riemannsche Flächen Abelsche Integrale , 1921 .

[32]  N. Chiarli.  Deficiency of linear series on the normalization of a space curve , 1984 .

[33]  Eben Matlis.  1-dimensional Cohen-Macaulay rings , 1973 .

[34]  Gerhard Pfister,et al.  Parallelization of Modular Algorithms , 2010, J. Symb. Comput..

[35]  N. Tschebotareff,et al.  Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören , 1926 .

[36]  M. Nöther,et al.  Ueber die algebraischen Functionen und ihre Anwendung in der Geometrie , 1874 .

[37]  J. Rafael Sendra,et al.  Parametrization of ε-rational curves: extended abstract , 2009, SNC '09.

[38]  B. L. Waerden,et al.  Einführung in die algebraische Geometrie , 1973 .

[39]  Algebraic Geometry I , 1994 .

[40]  J. Rafael Sendra,et al.  Rational Algebraic Curves , 2008 .

[41]  Qing Liu,et al.  Algebraic Geometry and Arithmetic Curves , 2002 .

[42]  Gert-Martin Greuel,et al.  Introduction to Singularities and Deformations , 2007 .

[43]  J. Milnor.  Singular points of complex hypersurfaces , 1968 .

[44]  Daniel Gorenstein,et al.  An arithmetic theory of adjoint plane curves , 1952 .

[45]  Claus Fieker,et al.  The use of bad primes in rational reconstruction , 2015, Math. Comput..

[46]  J. Stillwell,et al.  Plane Algebraic Curves , 1986 .

[47]  Gert-Martin Greuel,et al.  Normalization of rings , 2009, J. Symb. Comput..

[48]  Hlawka.  Einführung in die algebraische Geometrie , 1941 .

[49]  Mathias Schulze,et al.  Local Analysis of Grauert-Remmert-Type Normalization Algorithms , 2013, Int. J. Algebra Comput..

[50]  D. Rees.  TOPICS IN LOCAL ALGEBRA , 1969 .

[51]  Joe Harris,et al.  Geometry of Algebraic Curves: Volume I , 1984 .

[52]  S. Greco,et al.  On the theory of adjoints , 1979 .

[53]  A numerical criterion for simultaneous normalization , 2004, math/0408394.