Discrete behavior of Seshadri constants on surfaces

Working over , we show that, apart possibly from a unique limit point, the possible values of multi-point Seshadri constants for general points on smooth projective surfaces form a discrete set. In addition to its theoretical interest, this result is of practical value, which we demonstrate by giving significantly improved explicit lower bounds for Seshadri constants on and new results about ample divisors on blow ups of at general points.     

Fibred rational surfaces with extremal Mordell-Weil lattices

Slope inequalities are given for fibred rational surfaces according as the Clifford index of a general fibre. For fibred rational surfaces of Clifford index two, the Mordell-Weil lattices of maximal ranks are completely determined.Supported by The 21st Century COE Program named “Towards a new basic science: depth and synthesis”.     

Seshadri constants in a family of surfaces

The aim of this note is to study local and global Seshadri constants for a family of smooth surfaces with prescribed polarization. Received: 16 April 2001 / Published online: 26 April 2002     

Algebraic geometric codes on anticanonical surfaces

Algebraic geometric codes (or AG codes) provide a way to correct errors that occur during the transmission of digital information. AG codes on curves have been studied extensively, but much less work has been done for AG codes on higher dimensional varieties. In particular, we seek good bounds for the minimum distance.We study AG codes on anticanonical surfaces coming from blow-ups of P² at points on a line and points on the union of two lines. We can compute the dimension of such codes exactly due to known results. For certain families of these codes, we prove an exact result on the minimum distance. For other families, we obtain lower bounds on the minimum distance.     

Seshadri constants on surfaces of general type

We study Seshadri constants of the canonical bundle on minimal surfaces of general type. First, we prove that if the Seshadri constant ε(K _X , x) is between 0 and 1, then it is of the form (m − 1)/m for some integer m ≥ 2. Secondly, we study values of ε(K _X , x) for a very general point x and show that small values of the Seshadri constant are accounted for by the geometry of X.     

On uniruled degenerations of algebraic varieties with trivial canonical divisor

We give a uniruledness criterion on degenerate fibers in a family of varieties whose general fiber has a numerically trivial canonical divisor. The criterion is related with the so-called non-nef locus of the canonical divisor of the total space.      

Curves and 0-cycles on projective surfaces

LetC be a smooth curve with ag _n ¹ , i.e. a linear system of dimension 1 and degreen, lying on a smooth projective surfaceS. Let :S P ^N be the map associated to the line bundleK _S +[C] and letD be a general divisor of the given linear systemg _n ¹ . LetV be the linear space spanned by the image ofD through . We study the case in whichn:=dimV=1 and in general we discuss the case in whichn is small. The starting point is an analysis of the adjunction map using Bogomolov-Reider-Serrano techniques; several results from curve theory are also needed.     

Limits of Cartier divisors

Consider a one-parameter family of algebraic varieties degenerating to a reducible one. Our main result is a formula for the fundamental cycle of the limit subscheme of any family of effective Cartier divisors. The formula expresses this cycle as a sum of Cartier divisors, not necessarily effective, of the components of the limit variety.     

Cox rings of surfaces and the anticanonical Iitaka dimension

In this paper we investigate the relation between the finite generation of the Cox ring R(X) of a smooth projective surface X and its anticanonical Iitaka dimension κ(−KX).     

Seshadri constants in two half-periods

We calculate the exact value of the multiple point Seshadri constant in two halfperiods of an abelian surface with Picard number one. Received: 3 December 2003; revised: 9 July 2005     

Okounkov bodies and Seshadri constants

Okounkov bodies, which are closed convex sets defined for big line bundles, have rich information on the line bundles. On the other hand, Seshadri constants are invariants which measure the positivity of line bundles. In this paper, we prove that Okounkov bodies give lower bounds of Seshadri constants.     

Linear systems on a class of anticanonical rational threefolds

Let X be the blow-up of the three dimensional complex projective space along r   points in very general position on a smooth elliptic quartic curve B⊂P³$B\subset {\mathbb{P}}^3$ and let L∈Pic(X)$L\in \mathrm{Pic}(X)$ be any line bundle. The aim of this paper is to provide an explicit algorithm for determining the dimension of H⁰(X,L)$H^0(X,L)$.     

Seshadri constants and the generation of jets

In this paper we explore the connection between Seshadri constants and the generation of jets. It is well known that one way to view Seshadri constants is to consider them as measuring the rate of growth of the number of jets that multiples of a line bundle generate. Here we ask, conversely, what we can say about the number of jets once the Seshadri constant is known. As an application of our results, we prove a characterization of projective space among all Fano varieties in terms of Seshadri constants.     

Bielliptic surfaces as covers of rational surfaces

We present a construction of the bielliptic surfaces as covers of certain rational elliptic surfaces.     

An inequality for means with applications

We show that an almost trivial inequality between the first and second moment and the maximal value of a random variable can be used to slightly improve deep theorems. Received: 25 November 2006     

Calabi-Yau threefolds arising from fiber products of rational quasi-elliptic surfaces, II

We construct examples of supersingular Calabi-Yau threefolds in characteristic 2 making use of the method by Schoen. Unirational Calabi-Yau threefolds of five different topological types are obtained. There are two examples with the third Betti number zero among them, and they are counted as other examples of non-liftable Calabi-Yau threefolds in characteristic 2 after the one given by Schröer.