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A note on the ampleness of non-integral divisors  [PDF]
Stefano Urbinati
Mathematics , 2015,
Abstract: We investigate which of the characterizations of integral ample divisors can be extended to $\mathbb{Q}$-divisors and $\mathbb{R}$-divisors when associating to the Weil divisor the line bundle of its integral part.
Homotopy groups of complements to ample divisors  [PDF]
A. Libgober
Mathematics , 2004,
Abstract: We study the homotopy groups of complements to reducible divisors on non-singular projective varieties with ample components and isolated non normal crossings. We prove a vanishing theorem generalizing conditions for commutativity of the fundamental groups. The calculation of supports of non vanishing homotopy groups as modules over the fundamental group in terms of the geometry of the locus of non-normal crossings is discussed. We review previous work on the local study of isolated non-normal crossings and relate the motivic zeta function to the local polytopes of quasiadjunction. As an application, we obtain information about the support loci of homotopy groups of arrangements of hyperplanes
Ample divisors on the blow up of P^3 at points  [PDF]
Flavio Angelini
Mathematics , 1996,
Abstract: We give a condition for certain divisors on the blow up of P^3 at points in general position to be ample. The result extends a theorem of G. Xu on the blow up of the projective plane.
Constructing new ample divisors out of old ones  [PDF]
Paul Biran
Mathematics , 1997,
Abstract: We prove a gluing theorem which allows to construct an ample divisor on a rational surface from two given ample divisors on simpler surfaces. This theorem combined with the Cremona action on the ample cone gives rise to an algorithm for constructing new ample divisors. We then propose a conjecture relating continued fractions approximations and Seshadri-like constants of line bundles over rational surfaces. By applying our algorithm recursively we verify our conjecture in many cases and obtain new asymptotic estimates on these constants. Finally, we explain the intuition behind the gluing theorem in terms of symplectic geometry and propose generalizations.
Ample subvarieties and q-ample divisors  [PDF]
John Christian Ottem
Mathematics , 2011,
Abstract: We introduce a notion of ampleness for subschemes of higher codimension using the theory of q-ample line bundles. We also investigate certain geometric properties satisfied by ample subvarieties, e.g. the Lefschetz hyperplane theorems and numerical positivity. Using these properties, we also construct a counterexample to the converse of the Andreotti-Grauert vanishing theorem.
Weil divisors on rational normal scrolls  [PDF]
Rita Ferraro
Mathematics , 2001,
Abstract: The aim of this paper is to study Weil divisors on a singular rational normal scroll X. In particular the author describes explicitly the group of divisorial sheaves associated to Weil divisors on X, via the direct image of the Picard group of the canonical resolution of X, which is very well known. The analysis naturally splits in two cases, depending on the codimension of the vertex of X (>2 or =2). When the codimension of the vertex is >2, then the two groups are isomorphic and there is a natural intersection form in X inherited from the one in the canonical resolution. When the codimension of the vertex is 2 this is no longer true and to overcome this problem the author introduces the concept of integral total transform of a divisor in the resolution of X. In this case the author defines a (non-linear) intersection number of two divisors, that represents the degree of their scheme-theoretic intersection if they are effective with no common components. Lastly there are some examples and applications.
Cartier and Weil Divisors on Varieties with Quotient Singularities  [PDF]
Enrique Artal Bartolo,Jorge Martín-Morales,Jorge Ortigas-Galindo
Mathematics , 2011, DOI: 10.1142/S0129167X14501006
Abstract: The main goal of this paper is to show that the notions of Weil and Cartier $\mathbb{Q}$-divisors coincide for $V$-manifolds and give a procedure to express a rational Weil divisor as a rational Cartier divisor. The theory is illustrated on weighted projective spaces and weighted blow-ups.
A view on extending morphisms from ample divisors  [PDF]
Mauro C. Beltrametti,Paltin Ionescu
Mathematics , 2009,
Abstract: The philosophy that ``a projective manifold is more special than any of its smooth hyperplane sections" was one of the classical principles of projective geometry. Lefschetz type results and related vanishing theorems were among the typically used techniques. We shall survey most of the problems, results and conjectures in this area, using the modern setting of ample divisors, and (some aspects of) Mori theory.
Special Fano manifolds as ample divisors
Cristiana Sacchi
Le Matematiche , 1993,
Abstract: Let A be a projective manifold of dimension n at least 3 contained as an ample divisor in a projective manifold X and let L=OX(A). In this paper we study the pairs (X,L) in the following two cases: i) A a Fano manifold of coindex 3 and Picard number 1; A a Fano manifold of product type.
On Effective Non-vanishing of Weil Divisors on Algebraic Surfaces  [PDF]
Qihong Xie
Mathematics , 2004,
Abstract: We give a counterexample and some conclusions for effective non-vanishing of Weil divisors on algebraic surfaces.
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