Pencils and nets of small degree on curves on smooth, projective surfaces of Picard rank 1 and very ample generator

TL;DR

This paper investigates the geometric properties of curves on certain algebraic surfaces, demonstrating that specific linear systems are determined by hyperplane sections intersecting the surface in a controlled manner.

Contribution

It establishes a new link between linear systems on curves and multisecant hyperplanes on surfaces with Picard rank 1 and very ample generators.

Findings

Linear systems on curves are cut out by multisecant hyperplanes.

Results apply to curves with degree m ≥ 5.

Provides geometric characterization of linear systems on such surfaces.

Abstract

Let S be a smooth, projective surface of Picard rank 1 and very ample generator embedding S into P^n. Let C be a smooth curve in O(m) for m \geq 5. We prove that any base-point free, complete g^r_d on C for r\in\{1,2\} and d small enough is cut out by a hyperplane section restricted to a multisecant (n-r-1)-plane.