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