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Abstract. We consider smooth surfaces in P 4 and we prove that, under cerain hyphotheses, this surfaces actually contain a plane curve. Then we prove that the degree of such surfaces is bounded. This yields a result on codimension two smooth subcanonical subvarieties in P n, n ≥ 5 giving further evidence to Hartshorne conjecture in codimension two. This is a short summary of the contents of two papers, see [1] and [2]. We work over an algebraically closed field of characteristic zero. The main results are: THEOREM 1. Let � ⊂ P 4 be an hypersurface of degree s with a (s-2)-uple plane, then the degree of smooth surfaces S ⊂ � with q(S) = 0 is bounded. THEOREM 2. Let S ⊂ P 4 be a smooth surface with q(S) = 0 and lying on a quartic hypersurface �, such that Sing(�) has dimension two, then d = deg(S) ≤ 40. As an application to codim. two subvarieties in P n we have: THEOREM 3. Let X ⊂ P n, n ≥ 5, be a smooth codimension two subcanonicalsubvariety, lying on a hypersurface � of degree s having alinear subspace K of codimension two and multiplicity (s − 2).Then X is a complete intersection

Year: 2011

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