Algebraic surfaces of general type with P9 smaller or equal to 7 and K very ample
Let X ⊂ P6 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.
In this work we study the moduli part in the canonical bundle formula of an lc-trivial fibration whose general fibre is a rational curve. If is the Cartier index of the fibre, it was expected that would provide a bound on the denominators of the moduli part. Here we prove that such a bound cannot even be polynomial in , we provide a bound and an example where the smallest integer that clears the denominators of the moduli part is . Moreover we prove that even locally the denominators depend...