Weierstrass \(n\)-semigroups with even \(n\) and curves on toric surfaces. (English) Zbl 1466.14041
Summary: First, we give some Weierstrass semigroups which cannot be attained by any smooth curve on a smooth compact toric surface. Next, for any integer \(l\geq 2\) we describe the Weierstrass semigroup of a total ramification point of a cyclic covering of the projective line with degree \(l\) using \(l-1\) non-negative integers. And finally, we will study smooth curves \(C\) lying on a smooth compact toric surface \(S\) acted by the torus \(T\) which is a dense open subset of \(S\). For an even integer \(n\leq 10\) we characterize the Weierstrass semigroup of a total ramification point of a cyclic covering of degree \(n\) which is the restriction to \(C\) of a toric fibration of \(S\) such that the ramification point lies on some \(T\)-invariant divisor.
MSC:
| 14H55 | Riemann surfaces; Weierstrass points; gap sequences |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 14H30 | Coverings of curves, fundamental group |
| 14H50 | Plane and space curves |
| 14H51 | Special divisors on curves (gonality, Brill-Noether theory) |
Keywords:
toric surface; smooth curve; Weierstrass semigroup; plane curve; double cover; cyclic coveringReferences:
| [1] | Fulton, W., Introduction to Toric Varieties, Ann. of Math. Stud., vol. 131 (1993), Princeton U. Press: Princeton U. Press Princeton · Zbl 0813.14039 |
| [2] | Harui, T.; Komeda, J.; Ohbuchi, A., On double coverings between smooth plane curves, Kodai Math. J., 31, 257-262 (2008) · Zbl 1145.14027 |
| [3] | Kawaguchi, R., The gonality and the Clifford index of curves on a toric surface, J. Algebra, 449, 660-686 (2016) · Zbl 1405.14079 |
| [4] | Kawaguchi, R., Weierstrass gap sequences on curves on toric surfaces, Kodai Math. J., 33, 63-86 (2010) · Zbl 1221.14042 |
| [5] | Kawaguchi, R.; Komeda, J., Weierstrass semigroups satisfying MP equalities and curves on toric surfaces, Bull. Braz. Math. Soc, New Ser., 51, 107-123 (2020) · Zbl 1439.14111 |
| [6] | Kim, S. J.; Komeda, J., Numerical semigroups which cannot be realized as semigroups of Galois Weierstrass points, Arch. Math., 76, 265-273 (2001) · Zbl 1076.14518 |
| [7] | Kim, S. J.; Komeda, J., Weierstrass semigroups on double covers of plane curves of degree 5, Kodai Math. J., 38, 270-288 (2015) · Zbl 1327.14159 |
| [8] | Kim, S. J.; Komeda, J., Weierstrass semigroups on double covers of plane curves of degree six with total flexes, Bull. Korean Math. Soc., 55, 611-624 (2018) · Zbl 1401.14158 |
| [9] | Komeda, J., On Weierstrass points whose first non-gaps are four, J. Reine Angew. Math., 341, 68-86 (1983) · Zbl 0498.30053 |
| [10] | Komeda, J., On Weierstrass semigroups of double coverings of genus three curves, Semigroup Forum, 83, 479-488 (2011) · Zbl 1244.14025 |
| [11] | Komeda, J.; Ohbuchi, A., On double coverings of a pointed non-singular curve with any Weierstrass semigroup, Tsukuba J. Math., 31, 205-215 (2007) · Zbl 1154.14023 |
| [12] | Martens, G., On Curves on K3 Surfaces, Lecture Notes in Math., vol. 1389, 174-182 (1988), Springer-Verlag · Zbl 0698.14036 |
| [13] | Torres, F., Weierstrass points and double coverings of curves with application: symmetric numerical semigroups which cannot be realized as Weierstrass semigroups, Manuscr. Math., 83, 39-58 (1994) · Zbl 0838.14025 |
| [14] | Reid, M., Special linear systems on curves lying on a K3 surface, J. Lond. Math. Soc., 13, 454-458 (1976) · Zbl 0339.14021 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
