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Osculating Varieties of Veronese Varieties and Their Higher Secant Varieties

Published online by Cambridge University Press:  20 November 2018

A. Bernardi
Affiliation:
Dipartimento di Matematica di Bologna, Porta San Donato 5, 40126, Bologna, Italia email: abernardi@dm.unibo.it
M. V. Catalisano
Affiliation:
DIPTEM, Università di Genova, Genova, Italia email: catalisano@dimet.unige.it
A. Gimigliano
Affiliation:
Dipartimento di Matematica and C.I.R.A.M., Università di Bologna, Bologna, Italia email: gimiglia@dm.unibo.it
M. Idà
Affiliation:
Dipartimento di Matematica, Università di Bologna, Bologna, Italia email: ida@dm.unibo.it
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Abstract

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We consider the $k$-osculating varieties ${{O}_{k,\,n.d}}$ to the (Veronese) $d$-uple embeddings of ${{\mathbb{P}}^{n}}$. We study the dimension of their higher secant varieties via inverse systems (apolarity). By associating certain 0-dimensional schemes $Y\,\subset \,{{\mathbb{P}}^{n}}$ to $O_{k,n,d}^{s}$ and by studying their Hilbert functions, we are able, in several cases, to determine whether those secant varieties are defective or not.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2007

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