IJPAM: Volume 61, No. 4 (2010)

RANKS AND BORDER RANKS:
SOME INVARIANTS TO BOUND
THEM FOR VERONESE EMBEDDINGS

E. Ballico
Department of Mathematics
University of Trento
38 123 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: [email protected]


Abstract.Let $X_{m,d} \subset \mathbb {P}^{n_{m,d}}$, $n_{m,d}:= \binom{m+d}{m}-1$, be the Veronese embedding of order $d$ of $\mathbb {P}^m$. For any $P\in \mathbb {P}^{n_{m,d}}$ the $X_{m,d}$-rank $r_{X_{m,d}}(P)$ of $P$ is the minimal cardilaty of $S\subset X_{m,d}$ such that $P\in \langle S\rangle$. Fix a zero-dimensional scheme $Z\subset X_{m,d}$. Here we give a way to find upper bounds for all $r_{X_{m,d}}(P)$, $p\in \langle Z\rangle$ in terms of geometrical, numerical or cohomological properties of $Z$.

Received: February 22, 2010

AMS Subject Classification: 14N05

Key Words and Phrases: $X$-rank, ranks, border rank, Veronese embedding

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 61
Issue: 4