Weierstrass gap sequences on curves on toric surfaces

The gonality and the Clifford index of curves on a toric surface

“…Assume that there exists an oblique side which has no lattice points except for two end points, and denote by D the T -invariant divisor corresponds to this side. In this case, a point P = C ∩ D is a total ramification point of a gonality pencil on C, and moreover, P satisfies the assumption in Corollary 1.6 in [8]. Hence one can determine the Weierstrass gap sequence at P by moving the oblique side similarly to Fig.…”

“…By combining Theorem 1.3 with results in [8], we can compute Weierstrass gap sequences at ramification points (with high ramification indexes) of a gonality pencil. For example, in this section, we consider trigonal curves and provide a geometric interpretation of the structure of gap sequences at ramification points.…”

“…On the other hand, Theorem 1.3 and Fact 2.2 show that the fiber F of a trigonal covering from C to P 1 is F ∼ D 4 + 2D 5 + 3D 6 + D 7 , which implies that P is a total ramification point. By applying Corollary 1.6 in [8], we can to determine the gap sequence at P as {j | the line 3X + Y = 3 + j has a lattice point in Int C }.…”

“…In fact, gap sequences at such points are well studied and the classification of them has been already completed ( [3], [4], [7], [9]). However, by combining Corollary 1.4 with results in [8], we can compute gap sequences in a completely different way and propose a novel geometric interpretation of the reason why the difference between types of gap sequences occurs. This approach can be adapted not only to trigonal curves but also to curves of higher gonality.…”