The ideal of curves of genus 2 on rational normal scrolls

TL;DR

This paper classifies the types of rational normal scrolls associated with genus 2 curves embedded in projective space and describes their ideals in terms of linear systems g^1_2 and g^1_3.

Contribution

It provides a complete listing of rational normal scrolls related to genus 2 curves and describes their ideals as sums of specific scroll ideals.

Findings

Classified all rational normal scrolls from linear systems g^1_2 and g^1_3 on genus 2 curves.

Described the ideal of such curves as sums of ideals of associated scrolls.

Connected the geometry of linear systems to the algebraic ideals of embedded curves.

Abstract

Given a smooth curve of genus 2 embedded in P^(d-2) with a complete linear system of degree d>=6, we list all types of rational normal scrolls arising from linear systems g^1_2 and g^1_3 on C. Furthermore, we give a description of the ideal of such a curve of genus 2 embedded in P^(d-2) as a sum of the ideal of the two-dimensional scroll defined by the unique g^1_2 on C and the ideal of a three-dimensional scroll arising from a g^1_3 on C and not containing the scroll defined by the g^1_2 on C.