IJPAM: Volume 70, No. 1 (2011)

A BOUNDEDNESS RESULT FOR PFAFF FIELDS

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


Abstract. Fix positive integers $s, d, n$ with $n \ge s+1$. Let $W\subseteq \mathbb {P}^n$ be an integral and Gorenstein projective variety of dimension $s+1$ such that $\dim (\mbox{\rm Sing}(W)) \le s-1$. Fix $M, H\in \mbox{\rm Pic}(W)$ with $H$ ample. Here we prove the existence of an integer $x_0(H,d,M)$ with the following property. Fix any integer $x\ge x_0(H,d,M)$ and any integral $X\in \vert M\otimes H^{\otimes x}\vert$ such that $\dim (\mbox{\rm Sing} (X)) \le s-2$; then there is no non-zero Pfaff field $\Omega _X^s \to \mathcal {O}_X(d)$. In particular $X$ is not a solution of a rank $s$ and degree $d$ Pfaff field on $\mathbb {P}^n$ whose singular locus does not contain $X$.

Received: February 25, 2011

AMS Subject Classification: 37F75, 14J60, 58A17

Key Words and Phrases: Pfaff field

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 70
Issue: 1