A computer aided approach to codimension 2 subvarieties of \({\mathbb{P}}_ n\), n\(\geq 6\). (English) Zbl 0581.14035

Holme, Audun; Schneider, Michael

J. Reine Angew. Math. 357, 205-220 (1985).

The paper under review deals with the problem of whether there are smooth 2-codimensional subvarieties of the complex projective space \({\mathbb{P}}^ n\) with \(n\geq 6\) other than complete intersections, or equivalently, whether there are indecomposable rank 2 vector bundles on \({\mathbb{P}}^ n\) (n\(\geq 6)\). A special case of a well known conjecture of Hartshorne states that the answer to this question should be negative at least if \(n\geq 7\). The authors present some numerical evidence for a partial truth of this conjecture in codimension 2 and \(n\geq 6\). In particular, they prove that the conjecture is true as soon as the codimension is 2, \(n\geq 6\) and the degree d satisfies the inequality \(d<(n-1)(n+5)\). The computations also yield a list of values of d such that \(d\geq (n-1)(n+5)\) for which the conjecture is still true. For example, this is the case if \(n=6\) and \(d=55\). These results improve previous results due to Barth and Van de Ven (n\(\geq 6\) and \(d<(n+5)/4)\), Ran, or Ballico and Chiantini (n\(\geq 6\) and \(d<(n+2)^ 2/4)\).

Reviewer: L.Bădescu

MSC:

14M07	Low codimension problems in algebraic geometry
14M10	Complete intersections
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
14-04	Software, source code, etc. for problems pertaining to algebraic geometry

Keywords:

Hartshorne conjecture; smooth 2-codimensional subvarieties of the complex projective space; complete intersections; indecomposable rank 2 vector bundles