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Applicable Algebra in Engineering, Communication and Computing

Erschienen in:

01.11.2013 | Original Paper

Computational aspects of gonal maps and radical parametrization of curves

verfasst von: J. Schicho, F.-O. Schreyer, M. Weimann

Erschienen in: Applicable Algebra in Engineering, Communication and Computing | Ausgabe 5/2013

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Abstract

We develop in this article an algorithm that, given a projective curve \(C\), computes a gonal map, that is, a finite morphism from \(C\) to \(\mathbb P ^1\) of minimal degree. Our method is based on the computation of scrollar syzygies of canonical curves. We develop an improved version of our algorithm for curves with a unique gonal map and we discuss a characterization of such curves in terms of Betti numbers. Finally, we derive an efficient algorithm for radical parametrization of curves of gonality \(\le 4\).

Vorheriger Artikel Computational algebraic geometry and its applications

Nächster Artikel On the dual minimum distance and minimum weight of codes from a quotient of the Hermitian curve

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Take care that what we call a \(p^{th}\) syzygy corresponds to a \((p-1)^{th}\) syzygy in [8].

Arbarello, E., Cornalba, M.: Footnotes to a paper of Beniamino Segre. Math. Ann. 256, 341–362 (1981)MathSciNetCrossRefMATH

Abarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of Algebraic Curves, vol. I. Springer, Heidelberg (1985)CrossRef

Aprodu, M.: Remarks on syzygies of d-gonal curves. Math. Res. Lett. 12, 387–400 (2005)MathSciNetCrossRefMATH

Aprodu, M., Farkas, G.: The green conjecture for smooth curves lying on arbitrary K3 surfaces. Comp. Math. 147, 839–851 (2011)MathSciNetCrossRefMATH

Ballico, E., Keem, C., Park, S.: Double covering of curves. Proc. Am. Math. Soc. 132, 3153–3158 (2004)MathSciNetCrossRefMATH

Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comp. 24, 235–265 (1997)MathSciNetCrossRefMATH

Graf, H.C., Bothmer, V.: Scrollar syzygies of general canonical curves with genus \(\le 8\). Trans. AMS 359(2), 465–488 (2007)CrossRefMATH

Graf, H.C., Bothmer, V.: Generic syzygy scheme. J. Pure Appl. Algebra 208(3), 867–876 (2007)MathSciNetCrossRefMATH

Choi, Y., Kim, S.: Gonality and Clifford index of projective curves on ruled surfaces. Proc. Am. Math. Soc. 140(2), 393–402 (2012)MathSciNetCrossRefMATH

10.

Coppens, M.: One-dimensional linear systems of type II on smooth curves, Ph.D., Utrecht 1983 Thesis Utrecht (1983)

11.

Coppens, M., Martens, G.: Secant spaces and Clifford’s theorem. Compos. Math. 78, 193–212 (1991)MathSciNetMATH

12.

Coskun, I.: Surfaces of low degree containing a canonical curve, computational algebraic and analytic geometry, 57–70, Contemp. Math., 572, Am. Math. Soc., Providence, RI, (2012)

13.

Eisenbud, D.: Geometry of Syzygies, Graduate Texts in Mathematics 229. Springer, Berlin (2005)

14.

Eisenbud, D., Harris, J.: The Kodaira dimension of the moduli space of curves of genus \(\ge 23\). Invent. Math. 90(2), 359–387 (1987)MathSciNetCrossRefMATH

15.

Eisenbud, D., Hartshorne, R., Schreyer, F.-O.: Twenty points in \({\mathbb{P}}^3\), http://arxiv.org/abs/1212.1841

16.

Eisenbud, D., Lange, H., Martens, G., schreyer, F.O.: The Clifford dimension of an projective curve. Compos. Math. 72, 173–204 (1989)MathSciNetMATH

17.

de Graaf, W.A., Harrison, M., Pilnikova, J., Schicho, J.: A Lie algebra method for the parametrization of Severi-Brauer surfaces. J. Algebra 303(2), 514–529 (2006)MathSciNetCrossRefMATH

18.

Grayson, D., Stillman, M.: Macaulay2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/

19.

Green, M.: Koszul cohomology and the geometry of projective varieties. J. Differ. Geom. 19, 125–171 (1984)MATH

20.

Harrison, M.C.: Explicit solution by radicals, gonal maps and plane models of algebraic curves of genus 5 or 6, preprint arXiv:1103.4946 (2011)

21.

Hirschowitz, A., Ramanan, S.: New evidence for Green’s conjecture on syzygies of canonical curves. Ann. Sci. ENS (4) 31(2), 145–152 (1998)

22.

Kempf, G.: Curves of \(g^1d\)’s. Compos. Math. 55(2), 15–162 (1985)MathSciNet

23.

Kim, S.: On the Clifford sequence of a general k-gonal curve. Indag. Math. NS. 8, 209–216 (1997)CrossRefMATH

24.

Kim, S.: k-Gonal Curves, Trends in Mathematics, vol. 2. Information Center for Mathematical Sciences (1999)

25.

Nagata, M.: On Rational Surfaces I, Memoirs of the College of Science. University of Kyoto, Series A, vol. 32, no. 3 (1960)

26.

Noether, M.: Ober die invariante DarsteUung algebraischer Funktionen. Math. Ann. 17, 163–284 (1880)MathSciNetCrossRef

27.

Ohkouchi, M., Sakai, F.: The gonality of singular plane curves. Tokyo J. Math. 27, 137–147 (2004)MathSciNetCrossRefMATH

28.

Petri, K.: Uber die invariante Darstellung algebraischer Funktionen einer Variablen. Math. Ann. 88, 243–289 (1923)MathSciNetCrossRef

29.

Pirola, G.P., Schlesinger, E.: A curve algebraically but not rationally uniformized by radicals. J. Algebra 289(2), 412–420 (2005)MathSciNetCrossRefMATH

30.

Sagraloff, M.: Syzygies and special linear series of canonical curves of genus 9, PhD Thesis, Saarland University (2005). http://arxiv.org/abs/math/0605758

31.

Sakai, F.: On the gonality of singular plane curves. In: Proceedings of the Symposium on Algebraic Geometry at Sado, 2006, pp. 32–48 (2007)

32.

Schicho, J., Sevilla, D.: Radical parametrization of trigonal curves. Contemp. Math. 572, 221–231 (2012)MathSciNetCrossRef

33.

Schreyer, F.-O.: Syzygies of canonical curves and special linear series. Math. Ann. 275, 105–137 (1986)MathSciNetCrossRefMATH

34.

Schreyer, F.-O.: Green’s Conjecture for the General p-Gonal Curve of Large Genus, in Algebraic Curves and Projective Geometry. Springer Lecture Notes, vol. 1389, pp. 254–260 (1988)

35.

Schreyer, F.-O.: A standard basis approach to syzygies of canonical curves. J. Reine Angew. Math. 421, 83–123 (1991)MathSciNetMATH

36.

Sweedler, M.: Using Gröbner bases to determine the algebraic and transcendental nature of field extensions. AAECC 10, 66–75 (1993)MathSciNet

37.

Voisin, C.: Courbes tétragonales et cohomologie de Koszul. J. Reine Angew. Math. 387, 111–121 (1988)MathSciNetMATH

38.

Voisin, C.: Green’s generic syzygy conjecture for curves of even genus lying on a \(K_3\) surface. J. Eur. Math. Soc. 4, 363–404 (2002)MathSciNetCrossRefMATH

39.

Voisin, C.: Green’s canonical syzygy conjecture for generic curves of odd genus. Compos. Math. 141, 1163–1190 (2005)MathSciNetCrossRefMATH

Titel: Computational aspects of gonal maps and radical parametrization of curves
verfasst von: J. Schicho
F.-O. Schreyer
M. Weimann
Publikationsdatum: 01.11.2013
Verlag: Springer Berlin Heidelberg
Erschienen in: Applicable Algebra in Engineering, Communication and Computing / Ausgabe 5/2013
Print ISSN: 0938-1279
Elektronische ISSN: 1432-0622
DOI: https://doi.org/10.1007/s00200-013-0205-0

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