IJPAM: Volume 97, No. 2 (2014)

ON THE $X$-RANK OF A POINTS OF THE TANGENT
DEVELOPABLE OF A CURVE IN A PROJECTIVE SPACE

E. Ballico
Department of Mathematics
University of Trento
38 123 Povo (Trento) - Via Sommarive, 14, ITALY


Abstract. Let $X\subset \mathbb {P}^n$ be a smooth curve. For any $P\in \mathbb {P}^n$ the $X$-rank of $P$ is the minimal cardinality of a set $S\subset X$ such that $P\in \langle S\rangle$, where $\langle \ \rangle$ denote the linear span. Let $\tau (X)\subset \mathbb {P}^n$ be the tangent developable of $X$. We compute upper bounds for the $X$-rank of all $P\in \tau (X)$ or of the general $P\in \tau (X)$, mainly if $X$ is a canonically embedded curve. To do that we define some invariants for the pair $(X,\mathcal {O}_X(1))$ and compute them if $X$ is canonically embedded and either $X$ is a smooth plane curve or it has general moduli.

Received: August 6, 2014

AMS Subject Classification: 14N05, 14H52

Key Words and Phrases: tangential developable, $X$-rank, canonical model, smooth plane curve

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DOI: 10.12732/ijpam.v97i2.14 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 97
Issue: 2
Pages: 253 - 262


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