The Dimension of the Hilbert Scheme of Special Threefolds^#

^#Communicated by L. Ein.

The Hilbert scheme of 3-folds in ℙ ⁿ , n ≥  6 , that are scrolls over ℙ ² or over a smooth quadric surface Q  ⊂ ℙ ³ or that are quadric or cubic fibrations over ℙ ¹ is studied. All known such threefolds of degree 7  ≤ d ≤  11 are shown to correspond to smooth points of an irreducible component of their Hilbert scheme, whose dimension is computed.

Key Words:

• Good determinantal subschemes
• Hilbert schemes
• Special threefolds

1991 Mathematics Subject Classification:

• Primary 14J30, 14M07, 14N25
• Secondary 14N30

ACKNOWLEDGMENTS

Partially supported by MIUR of the Italian Government in the framework of the National Research Project (Cofin 2002) Geometria sulle Varietà Algebriche.

The material in this paper is, in part, based upon work supported by the National Science Foundation (NSF) under Grant No. 0125068. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.

Notes

^# Communicated by L. Ein.