Abstract
We describe the pole behaviour of the regular differentials of projective algebraic curves in terms of discrete invariants of the singular points.
Similar content being viewed by others
References
[B] V. Bayer,Semigroup of two irreducible algebroid plane curves, manuscripta math.39 (1985), 207–241.
[D1] F. Delgado de la Mata,The semigroup of values of a curve singularity with several branches, manuscripta math.59(1987), 347–374.
[D2] F. Delgado de la Mata,Gorenstein curves and symmetry of the semigroup of values, manuscripta math.61 (1988), 285–296.
[F] E.S. Freitas, “Curvas não-classicas de Gorenstein de gênero aritmético 3 e 4”, Tese de doutorado, IMPA, Rio de Janeiro 1992 (Announcement in Atas da XI Escola de Algebra, São Paulo 1990).
[Ga] V.M. Galkin,Zeta functions of some one-dimensional rings, Izv. Akad. Nauk. SSSR Ser. Mat.37 (1973), 3–19.
[G] A. Garcia,Semigroups associated to singular points of plane curves, J. reine angew. Math.336 (1982), 165–184.
[GL] A. Garcia and R.F. Lax,Weierstrass weight of Gorenstein singularities with one or two branches, Preprint.
[G1] B. Green,On the Riemann-Roch theorem for orders in the ring of valuation vectors of a function field, manuscripta math.60 (1988), 259–276.
[G2] B. Green,Functional equations for zeta-functions of non-Gorenstein orders in global fields, manuscripta math.64 (1989), 485–502.
[Ha] R. Harthshorne, “Algebraic Geometry”, Springer-Verlag, New York 1977.
[H] H. Hironaka,On the arithmetic genera and the effective genera of algebraic curves, Mem. Kyoto30 (1957), 177–195.
[HK] J. Herzog and H. Kunz, “Der kanonische Modul eines Cohen-Macaulay Rings”, Springer Lecture Notes in Math.238 (1971).
[HW] H. Hasse and E. Witt,Zyklische unverzweigte Erweiterungskörper vom Primzahlgrade p über einem algebraischen Funktionenkörper der Characteristik p, Monatsh. Math. Phys.43 (1936), 477–492.
[J] W.E. Jenner,On zeta-functions of number fields, Duke Math. J.36 (1969), 669–671.
[K1] H.I. Karakas,On Rosenlicht's generalization of Riemann-Roch theorem and generalized Weierstrass points, Arch. Math.27 (1976), 134–147.
[K2] H.I. Karakas,Application of generalized Weierstrass points: divisibility of divisor classes, J. reine angew. Math.299/300 (1978), 388–395.
[Ka] N. Katz,Une formule de congruence pour la fonction ζ, in: SGA 7 II (1967–1969), Springer Lecture Notes in Math.340, 401–438.
[K] E. Kunz,The value-semigroup of a one-dimensional Gorenstein ring, Proc. Amer. Math. Soc.25 (1970), 748–751.
[L] S. Lang, “Introduction to algebraic and abelian functions”, Addison-Wesley, Reading 1972.
[LW] R.F. Lax and C. Widland,Gap sequences at a singularity, Pac. J. Math.150 (1991), 107–115.
[M] Ju. I. Manin,The Hasse-Witt matrix of an algebraic curve, Amer. Math. Soc. Transl.45 (1965), 245–264.
[R1] P. Roquette,Über den Riemann-Rochschen Satz in Funktionenkörpern vom Transzendenzgrad 1, Math. Nachr.19 (1958), 375–404.
[R2] P. Roquette,Über den Singularitätsgrad von Teilringen in Funktionenkörpern, Math. Z.77 (1961), 228–240.
[R] M. Rosenlicht,Equivalence relations on algebraic curves, Ann. of Math. (2)56 (1952), 169–191.
[Sch] F.K. Schmidt,Analytische Zahlentheorie in Körpern der Charakteristik p, Math. Z.33 (1931), 1–32.
[S1] J.-P. Serre,Sur la topologie des variétés algébriques en caracteristic p, Symp. Int. Top. Alg., México City 1958, 24–53.
[S2] J.-P. Serre, “Groupes algébriques et corps de classes”, Hermann (1959), Paris.
[S3] J.-P. Serre, “Zeta andL functions, Arithmetical Algebraic Geometry”, Harper and Row, New York 1965, 82–92.
[St] H. Stichtenoth,Die Hasse-Witt-Invariante eines Kongruenzfunktionenkörpers, Arch. Math.33 (1979), 357–360.
[S] K.-O. Stöhr,On the moduli spaces of Gorenstein curves with symmetric Weierstrass semigroups, J. reine angew. Math., to appear.
[SV] K.-O. Stöhr and P. Viana,A study of Hasse-Witt matrices by local methods, Math. Z.200 (1989), 397–407.
[SV1] K.-O. Stöhr and J.F. Voloch,Weierstrass points and curves over finite fields, Proc. London Math. Soc. (3)52 (1986), 1–19.
[SV2] K.-O. Stöhr and J.F. Voloch,A formula for the Cartier operator on plane algebraic curves, J. reine angew. Math.377 (1987), 49–64.
[Wa] R. Waldi,Wertehalbgruppe and Singularität einer ebenen algebraischen Kurve, Dissertation, Regensburg, 1972.
[W] A. Weil,Zur algebraischen Theorie der algebraischen Funktionen, J. reine angew. Math.179 (1938), 129–133.
[WL] G. Widland and R. F. Lax,Weierstrass points on Gorenstein curves, Pac. J. Math.142 (1990), 197–208.
Author information
Authors and Affiliations
About this article
Cite this article
Stöhr, KO. On the poles of regular differentials of singular curves. Bol. Soc. Bras. Mat 24, 105–136 (1993). https://doi.org/10.1007/BF01231698
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01231698