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Gonality of non-Gorenstein curves of genus five. (English) Zbl 1308.14029
Summary: We establish sufficient conditions for some curves to be trigonal and derive from them that most of non-Gorenstein curves of genus five are so. Afterwards, we show that the gonality of such a curve ranges from 2 to 5. Gonality is understood within a broader context, i.e., the $$g_{d}^{1}$$ may possibly admit a base point and correspond to a torsion free sheaf of rank one instead of a line bundle. This study comes along with a thorough description of possible canonical models and kinds of singularities.

##### MSC:
 14H20 Singularities of curves, local rings 14H45 Special algebraic curves and curves of low genus 14H51 Special divisors on curves (gonality, Brill-Noether theory)
##### Keywords:
singular curve; non-Gorenstein curve; max Noether theorem
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##### References:
 [1] Barucci, V.; D’Anna, M.; Fröberg, R., Analyticallyunramifiedone-dimensional semilocal rings and their value semigroups, Journal of Pure and AppliedAlgebra, 147, 215-254, (2000) · Zbl 0963.13021 [2] Barucci, V.; Fröberg, R., One-dimensional almost Gorenstein rings, Journal of Algebra, 188, 418-442, (1997) · Zbl 0874.13018 [3] Behnke, K.; Christophersen, J.A., Hypersurface sections and obstructions (rational surface singularities), Comp. Math., 77, 233-268, (1991) · Zbl 0728.14034 [4] Buchweitz, R.-O., On deformations of monomial curves, Lec. Not. Math., 777, 205-220, (1980) · Zbl 0428.32016 [5] A. Contiero and K.-O. Sthöhr. Upper Bounds for the Dimension of Moduli Spaces of Curves with Symmetric Weierstrss Semigroups, arXiv 1211.2011v1. · Zbl 0047.14503 [6] Coppens, M., Free linear systems on integral Gorenstein curves, J. Algebra, 145, 209-218, (1992) · Zbl 0770.14002 [7] Eisenbud, D.; Koh, J.; Stillman, M., Determinantal equations for curves of high degree, Amer. J.Math., 110, 513-539, (1988) · Zbl 0681.14027 [8] Kleiman, S.L.; Martins, R.V., The canonical model of a singular curvee, Geometria Dedicata, 139, 139-166, (2009) · Zbl 1172.14019 [9] Lichtenbaum, S.; Schlessinger, M., The cotangent complex of a morphism, Trans. Am. Math. Soc., 128, 41-70, (1967) · Zbl 0156.27201 [10] Martins, R.V., On trigonal non-Gorenstein curves with zero maroni invariant, Journal of Algebra, 275, 453-470, (2004) · Zbl 1060.14036 [11] Martins, R.V., Trigonal non-Gorenstein curves, Journal of Pure and Applied Algebra, 209, 873-882, (2007) · Zbl 1108.14026 [12] Pinkham, H., deformations of algebraic varieties with G_{m}-action, Astérisque, 20, 1-131, (1974) · Zbl 0304.14006 [13] Rosenlicht, M., Equivalence relations on algebraic curves, Annals of Mathematics, 56, 169-191, (1952) · Zbl 0047.14503 [14] Serre, J.P., Groupes algébriques et corps de classes, (1959) · Zbl 0097.35604 [15] Stevens, J., The versal deformation of universal curve singularities, Abh. Math. Sem. Univ. Hamburg, 63, 197-213, (1993) · Zbl 0847.14014 [16] Stöhr, K.-O., On the poles of regular differentials of singular curves, Bull. Brazilian Math. Soc., 24, 105-135, (1993) · Zbl 0788.14020 [17] Stöhr, K.-O., Hyperelliptic Gorenstein curves, J. Pur. Appl. Algebra, 135, 93-105, (1999) · Zbl 0940.14018 [18] Rosa, R.; Stöhr, K.-O., Trigonal Gorenstein curves, J. Pur. Appl. Algebra, 174, 187-205, (2002) · Zbl 1059.14038
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