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Let X ⊂ P be a smooth irreducible projective threefold, and d its degree. In this paper we prove that there exists a constant β such that for all X containing a smooth ruled surface as hyperplane section and not contained in a fourfold of degree less than or equal to 15, d ≤ β. Under some more restrictive hypothesis we prove an analogous result for threefolds containing a smooth ruled surface as hyperplane section and contained in a fourfold of degree less than or equal to 15.
Let be a smooth irreducible non degenerate surface over the complex numbers, . We define the projective genus of , denoted by , as the geometric genus of the singular curve of the projection of from a general linear subspace of codimension four. Denote by the sectional genus of . In this paper we conjecture that the only surfaces for which are the del Pezzo surface in , in and a conic bundle of degree 5 in . We prove that for if , a non negative integer, then where for a...
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