IJPAM: Volume 10, No. 4 (2004)


E. Ballico
Department of Mathematics
University of Trento
380 50 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: ballico@science.unitn.it

Abstract.Fix integers $r \ge 2$, $d \ge 2$, an infinite set $I$, a complex Hilbert space $H$ with an orthonormal basis equipotent to $I$ , and a subset $S$ of the set of all hyperplanes of ${\bf {P}}^{r-1}$. Assume $2 \le \mbox{\rm card}(S) \le \mbox{\rm card}(I)$. Let $H(d)$ be the set of all degree $d$ complex valued polynomials on $H$. For every $s\in S$ fix a closed hyperplane $M_s$ of $H(d)$. Here we prove the existence of an $r$-dimensional linear subspace $A$ of $H(d)$ not contained in any $M_s$ and such that, setting $H_s:= {\bf {P}}(A\cap M_s)$, $\cup _{s\in S} H_s$ is the subset of ${\bf {P}}(A) = {\bf {P}}^{r-1}$ parametrizing the singular degree $d$ hypersurfaces with equations given by $A$.

Received: November 25, 2003

AMS Subject Classification: 30K05

Key Words and Phrases: complex Hilbert space, continuous homogeneous polynomial, singular hypersurface, infinite-dimensional complex projective space

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 10
Issue: 4