On Constraints Imposed by Independent Gonal Morphisms for a Curve

In this thesis, we explore the restrictions imposed on the genus of a smooth curve $X$ which possesses at least three independent gonal morphisms to $\Pp^1$. We will prove a sharp lower bound on the dimension of global sections given by the sum of the divisors for the gonal morphisms. This inequality will provide an upper bound on the genus of a curve with the described properties. By considering the birational image of $X$ in $\Pp^1 \times \Pp^1 \times \Pp^1$ under the product of three pairwise independent morphisms, we observe that the boundary case for the previously mentioned inequality is closely related to the case where the image of $X$ is contained in a type 1-1-1 surface. Motivated by this phenomenon, we examine the constraints on the arithmetic genus of an irreducible curve in $\Pp^1 \times \Pp^1 \times \Pp^1$ whose natural projections are pairwise independent and all have degree 7.