Curves in projective space

Abstract

This thesis is mainly concerned with classification of nonsingular projective space curves with an emphasis on the degree-genus pairs. In the first chapter, we present basic notions together with a very general notion of an abstract nonsingular curve associated with a function field, which is necessary to understand the problem clearly. Based on Nagata’s work [25], [26], [27], we show that every nonsingular abstract curve can be embedded in some P N and projected to P 3 so that the resulting image is birational to the curve in P N and still nonsingular. As genus is a birational invariant, despite the fact that degree depends on the projective embedding of a curve, curves in P 3 give the most general setting for classification of possible degree-genus pairs. The first notable attempt to classify nonsingular space curves is given in the works of Halphen [11], and Noether [28]. Trying to find valid bounds for the genus of such a curve depending upon its degree, Halphen stated a correct result for these bounds with a wrong claim of construction of such curves with prescribed degree-genus pairs on a cubic surface. The fault in the existence statement of Halphen’s work was corrected later by the works of Gruson, Peskine [9], [10], and Mori [21], which proved the existence of such curves on quartic surfaces. In Chapter 2, we present how the fault appearing in Halphen’s work has been corrected along the lines of Gruson, Peskine, and Mori’s work in addition to some trivial cases such as genus 0, 1, and 2 together with hyperelliptic, and canonical curves.