Algebra & Number Theory

On nonprimitive Weierstrass points

Nathan Pflueger

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We give an upper bound for the codimension in g,1 of the variety G,1S of marked curves (C,p) with a given Weierstrass semigroup. The bound is a combinatorial quantity which we call the effective weight of the semigroup; it is a refinement of the weight of the semigroup, and differs from the weight precisely when the semigroup is not primitive. We prove that whenever the effective weight is less than g, the variety G,1S is nonempty and has a component of the predicted codimension. These results extend previous results of Eisenbud, Harris, and Komeda to the case of nonprimitive semigroups. We also survey other cases where the codimension of G,1S is known, as evidence that the effective weight estimate is correct in wider circumstances.

Article information

Algebra Number Theory, Volume 12, Number 8 (2018), 1923-1947.

Received: 2 September 2016
Revised: 5 January 2018
Accepted: 10 March 2018
First available in Project Euclid: 21 December 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H55: Riemann surfaces; Weierstrass points; gap sequences [See also 30Fxx]

Weierstrass points numerical semigroups algebraic curves limit linear series


Pflueger, Nathan. On nonprimitive Weierstrass points. Algebra Number Theory 12 (2018), no. 8, 1923--1947. doi:10.2140/ant.2018.12.1923.

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  • O. Amini and M. Baker, “Linear series on metrized complexes of algebraic curves”, Math. Ann. 362:1-2 (2015), 55–106.
  • E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves, I, Grundlehren der Math. Wissenschaften 267, Springer, 1985.
  • E. Arbarello, M. Cornalba, and P. A. Griffiths, Geometry of algebraic curves, II, Grundlehren der Math. Wissenschaften 268, Springer, 2011.
  • M. Bras-Amorós and S. Bulygin, “Towards a better understanding of the semigroup tree”, Semigroup Forum 79:3 (2009), 561–574.
  • E. M. Bullock, “Subcanonical points on algebraic curves”, Trans. Amer. Math. Soc. 365:1 (2013), 99–122.
  • E. M. Bullock, “Irreducibility and stable rationality of the loci of curves of genus at most six with a marked Weierstrass point”, Proc. Amer. Math. Soc. 142:4 (2014), 1121–1132.
  • M. Coppens and T. Kato, “Weierstrass gap sequence at total inflection points of nodal plane curves”, Tsukuba J. Math. 18:1 (1994), 119–129.
  • D. A. Cox, J. B. Little, and H. K. Schenck, Toric varieties, Graduate Studies in Math. 124, Amer. Math. Soc., Providence, RI, 2011.
  • D. Eisenbud and J. Harris, “Limit linear series: basic theory”, Invent. Math. 85:2 (1986), 337–371.
  • D. Eisenbud and J. Harris, “Existence, decomposition, and limits of certain Weierstrass points”, Invent. Math. 87:3 (1987), 495–515.
  • J. Harris, “On the Severi problem”, Invent. Math. 84:3 (1986), 445–461.
  • J. Harris and I. Morrison, Moduli of curves, Graduate Texts in Math. 187, Springer, 1998.
  • A. Hurwitz, “Ueber algebraische Gebilde mit eindeutigen Transformationen in sich”, Math. Ann. 41:3 (1892), 403–442.
  • N. Kaplan and L. Ye, “The proportion of Weierstrass semigroups”, J. Algebra 373 (2013), 377–391.
  • J. Komeda, “On primitive Schubert indices of genus $g$ and weight $g-1$”, J. Math. Soc. Japan 43:3 (1991), 437–445.
  • M. Kontsevich and A. Zorich, “Connected components of the moduli spaces of abelian differentials with prescribed singularities”, Invent. Math. 153:3 (2003), 631–678.
  • T. Nakano, “On the moduli space of pointed algebraic curves of low genus, II: Rationality”, Tokyo J. Math. 31:1 (2008), 147–160.
  • B. Osserman, “A limit linear series moduli scheme”, Ann. Inst. Fourier $($Grenoble$)$ 56:4 (2006), 1165–1205.
  • B. Osserman, “Limit linear series”, draft monograph, 2013,
  • B. Osserman, “Limit linear series for curves not of compact type”, preprint, 2014. To appear in J. Reine Angew. Math.
  • B. Osserman, “Limit linear series moduli stacks in higher rank”, preprint, 2014.
  • N. Pflueger, “Weierstrass semigroups on Castelnuovo curves”, preprint, 2016.
  • H. C. Pinkham, Deformations of algebraic varieties with $G_m$ action, Astérisque 20, Soc. Math. France, Paris, 1974.
  • Z. Ran, “Families of plane curves and their limits: Enriques' conjecture and beyond”, Ann. of Math. $(2)$ 130:1 (1989), 121–157.
  • D. S. Rim and M. A. Vitulli, “Weierstrass points and monomial curves”, J. Algebra 48:2 (1977), 454–476.
  • P. Deligne, “Quadriques”, pp. 62–81 in Groupes de monodromie en géométrie algébrique, II: Exposés X–XXII (Séminaire de Géométrie Algébrique du Bois Marie 1967–1969), edited by P. Deligne and N. Katz, Lecture Notes in Math. 340, Springer, 1973.
  • A. Zhai, “Fibonacci-like growth of numerical semigroups of a given genus”, Semigroup Forum 86:3 (2013), 634–662.