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Classification of Algebraic Space Curves, III

In: Algebraic Geometry and its Applications

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  • Robin Hartshorne

Abstract

Over a fixed algebraically closed ground field k, we consider irreducible non-singular projective curves in the projective space P k 3 . The original classification problem for algebraic space curves could be described as finding all such curves, giving their numerical invariants, and determining the algebraic families they belong to. This was the problem tackled in the great treatises of M. Noether [10] and G. Halphen [5] over a hundred years ago, and has been the subject of numerous investigations since then. The determination of which pairs can occur as the degree and genus of a non-singular space curve was stated by Halphen, but only properly proved within the last decade by Gruson and Peskine [4]. Now it is reasonable to ask for finer numerical data, for example one can ask for the postulation of all possible curves, which means for each n, the number of conditions for a hypersurface of degree n to contain the curve.

Suggested Citation

  • Robin Hartshorne, 1994. "Classification of Algebraic Space Curves, III," Springer Books, in: Chandrajit L. Bajaj (ed.), Algebraic Geometry and its Applications, chapter 5, pages 113-120, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-2628-4_5
    DOI: 10.1007/978-1-4612-2628-4_5
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