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:
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