Threefolds of degree 11 in P⁵

Summary

Introduction. This article is devoted to the classification of smooth projective threefolds in P⁵.

In [BSS] we classified degree 9 and 10 threefolds in P⁵; the lower degree varieties had already been classified (see [11], [12], [13], [O1], [O2]). In that article, we used known constraints and new results from adjunction theory ([BBS], [S1], [S5], [SV]) to restrict the possible invariants. We then used liaison to construct examples with the possible invariants. Uniqueness of the examples satisfying the invariants was also shown.

In this paper we extend the methods of [BSS] to deal with the degree 11 case. The list we obtain of the possible invariants is again short, and we have examples for every possible set of invariants. We refer the reader to the start of § 4 where there is a table giving the degree 11 classification. For the reader's convenience we have given a one page appendix to this paper with a table giving the known classification of degree ≤ 10 threefolds in P⁵.

Degree 11 is especially interesting because our calculations show that the number of possible sets of invariants begins to increase quite fast from degree 12 on. This is discussed in (4.6).

We would like to thank the DFG-Schwerpunktprogram “Komplexe Mannigfaltigkeiten” for making it possible for us to work together at the University of Bayreuth in the summer of 1988. The third author would like to thank the National Science Foundation (DMS 87-22330 and DMS 89-21702). The first and the third author would like to thank the University of Notre Dame for its support. We would like to thank Ms. Cinzia Matrl for the excellent typing.