IJPAM: Volume 82, No. 5 (2013)

YET ANOTHER $X$-RANK
CHARACTERIZATION OF RATIONAL NORMAL CURVES

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


Abstract. Fix positive integers $s, k_i$, $1\le i\le s$, such that $k_1 \ge 2$ and $2k < n$, where $k:= k_1+\cdots +k_s$. Let $X\subset \mathbb {P}^n$ be an integral and non-degenerate curve. For any $P\in \mathbb {P}^n$ the $X$-rank $r_X(P)$ of $P$ is the minimal cardinality of a set $S\subset X$ such that $P\in \langle S\rangle$. We prove that $X$ is not a rational normal curve if and only if the following condition holds: fix $s$ general points $P_1,\dots ,P_s\in X_{reg}$ and set $Z:= \sum _{i=1}^{s} k_iP_i$; then there is some $P\in \langle Z\rangle$ such that $P\notin \langle Z'\rangle$ for any $Z'\subsetneq Z$ and $r_X(P) \le n+1-k$.

Moreover, if $X$ is not a rational normal curve and we fix a finite set $E\subset X$, then we may find a set $S\subset X\setminus E$ with $\sharp (S) \le n+1-k$ and $P\in \langle S\rangle$.

Received: December 14, 2012

AMS Subject Classification: 14H50, 14N05

Key Words and Phrases: $X$-rank, rational normal curve

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DOI: 10.12732/ijpam.v82i5.10 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: 2013
Volume: 82
Issue: 5