IJPAM: Volume 91, No. 3 (2014)

ON THE GENERIC RANK OF
LINEAR SPANS OF TANGENT VECTORS

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


Abstract. Let $X\subset \mathbb {P}^r$ be an integral and non-degenerate variety. For each $P\in \mathbb {P}^r$ the $X$-rank $r_X(P)$ is the minimal cardinality of a set of $X$ whose linear span contains $P$. For each $O\in X_{reg}$ let $\alpha (X,O)$ be the maximal integer $r_X(P)$ for some $P$ in the tangent space of $X$ at $O$. Let $\alpha (X)_{gen}$ be the integer $\alpha (X,O)$ for a general $O\in X$. Let $\beta (X)$ be the maximum of all $\alpha (X,O)$, $O\in X_{reg}$. The integer $\alpha (X)_{gen}$ is useful to get an upper bound for the integers $r_X(P)$, $P\in \mathbb {P}^r$. We prove that $\alpha (X)_{gen} =\beta (X)$ when $X$ is the degree $d \ge 4$ Veronese embedding of a cubic hypersurface.

Received: November 17, 2013

AMS Subject Classification: 14N05, 14Q05, 15A69

Key Words and Phrases: $X$-rank, cubic hypersurface, tangent developable, generic tangent space

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DOI: 10.12732/ijpam.v91i3.8 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: 91
Issue: 3