# zbMATH — the first resource for mathematics

Weierstrass gap sequences and moduli varieties of trigonal curves. (English) Zbl 0768.14016
The Weierstrass gap sequence at an unramified point of a trigonal curve is composed by 2 sequences of consecutive integers; such a result was established by S. J. Kim [J. Pure Appl. Algebra 63, No. 2, 171-180 (1990; Zbl 0712.14019)]. With this in mind, the authors obtain the following main result: Theorem: Let $$g$$, $$\sigma$$ and $$\rho$$ be integers with $$g\geq 5$$ and $$\sigma<g<\rho<2\sigma+2$$. Then the moduli space of pointed trigonal curves of genus $$g$$ and Weierstrass gap sequence $$1,\dots,\sigma$$, $$\sigma+\rho-g+1,\dots,\rho$$ has dimension $$2g+3- \rho+\sigma$$ provided $$\rho<3\left[{g+1\over 2}\right]+4-\sigma$$. In the course of the proof of this result, the authors provide another proof of Kim’s result mentioned above.

##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences
##### Keywords:
Weierstrass gap sequence; trigonal curve
Full Text:
##### References:
 [1] Andreotti, A.; Mayer, A., On period relations for abelian integrals on algebraic curves, Ann. scuola norm. sup. Pisa, 21, 189-238, (1967) · Zbl 0222.14024 [2] Arbarello, A.; Cornalba, M.; Griffiths, P.; Harris, J., Geometry of algebraic curves, (1985), Springer New York · Zbl 0559.14017 [3] Canuto, G., Weierstrass points on trigonal curves of genus five, Bull. soc. math. France, 113, 157-182, (1985) · Zbl 0599.14024 [4] Coppens, M., The Weierstrass gap sequences of the total ramification points of trigonal coverings of P1, Indag. math., 47, 245-276, (1985) · Zbl 0592.14025 [5] Coppens, M., The Weierstrass gap sequences of the ordinary ramification points of trigonal coverings of $$P$$ existence of a kind of Weierstrass gap sequence, J. pure appl. algebra, 43, 11-25, (1986) · Zbl 0616.14012 [6] Kato, T.; Horiuchi, R., Weierstrass gap sequences at the ramification points of a trigonal Riemann surface, J. pure appl. algebra, 50, 271-285, (1988) · Zbl 0649.14009 [7] Kim, S., On the existence of Weierstrass gap sequences on trigonal curves, J. pure appl. algebra, 63, 171-180, (1990) · Zbl 0712.14019 [8] Hartshorne, R., Algebraic geometry, (1977), Springer New York · Zbl 0367.14001 [9] Maroni, A., Le serie lineare speciali sulle curve trigonali, Ann. mat. pura appl., 25, 341-354, (1946) · Zbl 0061.35407 [10] Oliveira, G., Weierstrass semigroups and the canonical ideal of non-trigonal curves, Manuscripta math., 71, 431-450, (1991) · Zbl 0742.14029 [11] Pinkham, H., Deformations of algebraic varieties with Gm-action, Astérisque, 20, 3-127, (1974) [12] Stöhr, K.-O.; Viana, P., A variant of Petri’s analysis of the canonical ideal of an algebraic curve, Manuscripta math., 61, 223-248, (1988) · Zbl 0661.14025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.