International Journal of Mathematics

Let (X, L) be a polarized manifold of dimension 3. In this paper, we consider a lower bound for h⁰(K_X + 2L). We prove that h⁰(K_X + 2L) > 0 if K_X + 2L is nef, which is a conjecture of Beltrametti–Sommese for polarized 3-folds. Moreover we classify polarized 3-folds (X, L) with h⁰(K_X + 2L) = 1 under the assumption that K_X + 2L is nef.

We show that the method to construct C*-algebras from topological graphs, introduced in our previous paper, generalizes many known constructions. We give many ways to make new topological graphs from old ones, and study the relation of C*-algebras constructed from them. We also give a characterization of our C*-algebras in terms of their representation theory.

We study families of ropes of any codimension that are supported on lines. We construct suitable smooth parameter spaces and conclude that all ropes of fixed degree and genus lie in the same component of the corresponding Hilbert scheme. We show that this component is generically smooth if the genus is small enough unless the characteristic of the ground field is two and the curves under consideration have degree two. In this case the component is non-reduced.