Adjoint divisors on algebraic curves

Let C be a numerically connected curve lying on a smooth algebraic surface. We show that if is an ample invertible sheaf satisfying some technical numerical hypotheses then is normally generated. As a corollary we show that the sheaf ω_C^⊗2 on a numerically connected curve C of arithmetic genus p_a?3 is normally generated if ω_C is ample and does not exist a subcurve B⊂C such that p_a(B)=1=B(C−B).     

Numerically positive divisors on algebraic surfaces

Numerically positive line bundles on a complex projective smooth algebraic surfaceS are studied. In particular for any such line bundleL Pic(S) we prove the following facts: (i)g(L) 0 and (ii)L is ample ifg(L) 1,g standing for the arithmetic genus. Some applications are discussed. We also investigate numerically positive non-ample line bundlesL withg(L)=2.     

Even points on an algebraic curve

We show that the points of a global function field, whose classes are 2-divisible in the Picard group, form a connected regular infinite graph, with the incidence relation generalizing the well known quadratic reciprocity law. We prove that for every global function field the diameter of this graph is precisely 2. In addition we develop an analog of global square theorem that concerns points 2-divisible in the Picard group.     

Homological equivalence relations on an algebraic curve

We study the collection of homological equivalence relations on a fixed curve. We construct a moduli space for pairs consisting of a curve of genus g and a homological equivalence relation of degree n on the curve, and a classifying set for homological equivalence relations of degree n on a fixed curve, modulo automorphisms of the curve. We identify a special type of homological equivalence relations, and we characterize the special homological equivalence relations in terms of the existence of elliptic curves in the Jacobian of the curve.     

Estimation of a sum along an algebraic curve

Let be an algebraic curve determined over a finite field k = [q]; e,x are subsidiary additive and multiplicative characters of the field k;, are functions in determined over k and satisfying some natural conditions. If P passes through the points of curve , rational over k, then where constant C depends only on the powers of ,,.Translated from Matematicheskie Zametki, Vol. 5, No. 3, pp. 373–380, March, 1969.     

On the gonality sequence of an algebraic curve

For any smooth irreducible projective curve X, the gonality sequence ${\{d_r \;| \; r \in \mathbb N\}}$ is a strictly increasing sequence of positive integer invariants of X. In most known cases d _r+1 is not much bigger than d _r . In our terminology this means the numbers d _r satisfy the slope inequality. It is the aim of this paper to study cases when this is not true. We give examples for this of extremal curves in ${{\mathbb P}^r}$ , for curves on a general K3-surface in ${{\mathbb P}^r}$ and for complete intersections in ${{\mathbb P}^3}$ .     

The algebraic independence of the sum of divisors functions

Let σj(n)=_∑d|ndj be the sum of divisors function, and let I be the identity function. When considering only one input variable n, we show that the set of functions is algebraically independent. With two input variables, we give a non-trivial identity involving the sum of divisors function, prove its uniqueness, and use it to prove that any perfect number n must have the form n=rσ(r)/(2r−σ(r)), with some restrictions on r. This generalizes the known forms for both even and odd perfect numbers.     

Equivariant completions of homogenous algebraic varieties by homogenous divisors

Complete smooth complex algebraic varieties with an almost transitive action of a linear algebraic group are studied. They are classified in the case, when the complement of the open orbit is a homogeneous hypersurface. If the group and the isotropy subgroup at a generic point are both reductive, then there exists a natural one-to-one correspondence between these two-orbit varieties and compact riemannian symmetric spaces of rank one.     

Linear algebra methods for forbidden configurations

We say a matrix is simple if it is a (0,1)-matrix with no repeated columns. Given m and a k×l (0,1)-matrix F we define forb(m,F) as the maximum number of columns in a simple m-rowed matrix A for which no k×l submatrix of A is a row and column permutation of F. In set theory notation, F is a forbidden trace. For all k-rowed F (simple or non-simple) Füredi has shown that forb(m,F) is O(m ^k ). We are able to determine for which k-rowed F we have that forb(m,F) is O(m ^k−1) and for which k-rowed F we have that forb(m,F) is Θ(m ^k ).