Remarks on a conjecture of Hartshorne (Q1813038)

We study the following conjecture (stated by R. Hartshorne): Let X and Y be two smooth subvarieties of a complex projective variety Z. Assume \(\dim (X)+\dim (Y)\geq \dim (Z)\) and that the normal bundles of X and Y are ample. Then \(X\cap Y\neq 0.\) This conjectures is easily seen to be true in case one of the subvarieties has codimension 1. Thus the first non-trivial case is when \(\dim (X)=\dim (Y)=2\) and \(\dim (Z)=1.\) The methods used here involve the reduction to positive characteristic and the use of the Frobenius operation. - We assume that the Néron- Severi groups of Z and of almost all its reductions mod p be \({\mathbb{Z}}\) (a condition which is satisfied if for instance Z is a complete intersection in \({\mathbb{P}}^ n\) or if \(b_ 2(Z)=1)\), and prove that the conjecture is true whenever the normal bundle N of one subvariety, say X, satisfies a further numerical condition. This is if the form \(c^ 2_ 1(N)-2c_ 2(N)>C\), where C is a number depending on the Chern classes of X. This number is explicitly computed for many classes of surfaces.