Weierstrass semigroups satisfying MP equalities and curves on toric surfaces. (English) Zbl 1439.14111
Summary: A numerical semigroup \(H\) is said to be cyclic if it is the Weierstrass semigroup of a total ramification point of some cyclic covering of the projective line. In this case, the elements of the standard basis of \(H\) satisfy numerical conditions that we have chosen to term MP equalities, after Morrison and Pinkham who first proved them. The converse is not true, as there are semigroups satisfying the MP equalities that are not cyclic. In this paper, we consider the situation of a smooth curve \(C\) lying on a smooth compact toric surface \(S\) acted on by the torus \(T\) which is a dense open subset of \(S\). We prove that the Weierstrass semigroup of a total ramification point of the restriction to \(C\) of a toric fibration of \(S\), which lies on some \(T\)-invariant divisor, is cyclic if and only if it satisfies the MP equalities.
MSC:
| 14H55 | Riemann surfaces; Weierstrass points; gap sequences |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 14H30 | Coverings of curves, fundamental group |
| 14H51 | Special divisors on curves (gonality, Brill-Noether theory) |
| 20M14 | Commutative semigroups |
Keywords:
toric surface; smooth curve; Weierstrass semigroup; numerical semigroup; MP equalities; cyclic coveringReferences:
| [1] | Fulton, W., Introduction to Toric Varieties, Annals of Mathematics Studies 131 (1993), Princeton: Princeton University Press, Princeton · Zbl 0813.14039 |
| [2] | Kawaguchi, R., Weierstrass gap sequences on curves on toric surfaces, Kodai Math. J., 33, 63-86 (2010) · Zbl 1221.14042 · doi:10.2996/kmj/1270559158 |
| [3] | Kawaguchi, R., The gonality and the Clifford index of curves on a toric surface, J. Algebra, 449, 660-686 (2016) · Zbl 1405.14079 · doi:10.1016/j.jalgebra.2015.11.017 |
| [4] | Kim, Sj; Komeda, J., Numerical semigroups which cannot be realized as semigroups of Galois Weierstrass points, Arch. Math., 76, 265-273 (2001) · Zbl 1076.14518 · doi:10.1007/s000130050568 |
| [5] | Kim, Sj; Komeda, J., The Weierstrass semigroup of a pair of Galois Weierstrass points with prime degree on a curve, Bull. Braz. Math. Soc. New Ser., 36, 127-142 (2005) · Zbl 1082.14036 · doi:10.1007/s00574-005-0032-4 |
| [6] | Morrison, I.; Pinkham, H., Galois Weierstrass points and Hurwitz characters, Ann. Math., 124, 591-625 (1986) · Zbl 0624.14018 · doi:10.2307/2007094 |
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