Abstract

We define the C -rank associated to a projective curve and describe the strata of points having constant rank.

Highlights

  • Let Y ⊂ Pn be a nondegenerate variety

  • If Pn = P(V1 ⊗ · · · ⊗ Vr ) is the projective space associated to the tensor product of r vector spaces and Y is the variety of decomposable tensors, the Y -rank is called the tensor rank

  • If Pn = P(Sm,d) is the projective space associated to the vector space of degree d homogeneous forms in m variables, and Y is the locus of forms that are the dth power of a linear form the Y -rank of a form is called the rank of the form

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Summary

Introduction

Let Y ⊂ Pn be a nondegenerate variety. Let x ∈ Pn be a point, we define the Y -rank of x as the smallest number r such that x is in the linear span of r points in Y. A point in a tangent line will have in general greater rank, it is a limit of rank two points This behaviour is already present in the case of the rational normal curve. The only forms having rank d are those on tangent lines to the rational normal curve, that is, forms that can be written as P = Ld1−1L2 with L1 and L2 different (up to scalar) linear forms. If C is immersed in Pn by the complete linear system associated to a degree d divisor (d ≥ 4g + 1), a point in a tangent line to C (different from the tangency point) has rank n − g, it is a limit of rank 2 points (Theorem 2).

Bounds for the rank
Linear spaces spanned by divisor of a curve
Proofs of the main theorems
Rational normal curves
Elliptic normal curves
Even n
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