Remarks on the defining equations of smooth threefolds in P-5

Abstract

In this paper, we prove the vanishing of the first cohomology groups of ideal sheaves twisted by small integers for smooth threefolds in P-5 defined over the complex number field C by using Le Potier vanishing theorem and Castelnuovo regularity for vector bundles, which contributes Peskine-Van de Ven conjecture for 2-normality.