IJPAM: Volume 55, No. 4 (2009)

INTEGRABLE FORMS AND UNIONS
OF VARIETIES OF MINIMAL DEGREES

E. Ballico
Department of Mathematics
University of Trento
38 123 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: ballico@science.unitn.it


Abstract.Let $W$ be a finite dimensional subspace of the set of all holomorphic $1$-form at $(\mathbb {C}^n,0)$ and $I_W\subseteq \mathbb {P}(W)$ the subset of $W$ formed by the integrable $1$-forms. Assume $\mbox{\rm rank}(W) = \dim (W)$. Let $Y_i$, $1 \le s$, be some non-linear irreducible components of $I_W$ such that $Y_1\cup \cdots \cup Y_s$ is connected. Here we use the case $s=1$ recently proved by J.V. Pereira and C. Perrone to prove that $Y_1\cup \cdots \cup Y_s$ has minimal degree in its linear span.

Received: August 16, 2009

AMS Subject Classification: 32S65

Key Words and Phrases: integrable 1-forms, holomorphic foliations, rational normal curves

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 55
Issue: 4