Consider a finite morphism f : X → Y of smooth, projective varieties over a finite field F. Suppose X is the vanishing locus in PN of r forms of degree at most d. We show that there is a constant C depending only on (N, r, d) and deg( f ) such that if |F| > C, then f (F) : X(F) → Y(F) is injective if and only if it is surjective.

We describe V G k for general groups in §2. These formulas were first written down by E. Verlinde [23] in the context of Conformal Field theory. The interest towards them in Algebraic Geometry stems form the fact that they give the Hilbert function of moduli spaces of principle bundles over projective curves. More precisely, let C be a smooth projective curve of genus g, and let MC be the modul...

Using an Euclidean approach, we prove a new upper bound for the number of closed points of degree 2 on a smooth absolutely irreducible projective algebraic curve defined over the finite field Fq. This bound enables us to provide explicit conditions on q, g and π for the nonexistence of absolutely irreducible projective algebraic curves defined over Fq of geometric genus g, arithmetic genus π an...

Let C be a smooth projective irreducible curve defined over a finite field Fq and K = Fq(C). We show that every non-torsion element α ∈ K of degree d over K of a Drinfeld A-module φ defined over a finite extension L of K has canonical height ĥφ(α) at least c/d, where c is a constant depending only on φ. Similarly, if E/K is a non-constant elliptic curve defined over a function field K = l(C) of...

In [2, Section 1.6] Peskine and Szpiro prove a theorem on adic approximations of finite free resolutions over local rings which, together with M. Artin's Approximation Theorem [1], allows them to "descend" modules of finite projective dimension over the completions of certain local rings to modules of finite projective dimension over finite etale extensions of those rings. In this note we will ...

In this paper we consider some finite generalized polygons, defined over a field with characteristic 2, that admit an embedding in a projective or affine space over a field with characteristic unequal to 2. In particular, we classify the (lax) embeddings of the unique generalized quadrangle H(3, 4) of order (4, 2). We also classify all (lax) embeddings of both the split Cayley hexagon H(2) and ...

Let k be a perfect field, and X an irreducible smooth projective curve over k. We give criterion for vector bundle to admit logarithmic connection singular finite subset of with given residues, where residues are assumed rigid.

Let F/K be an algebraic function field in one variable over a finite field of constants K, i.e., F is a finite algebraic extension of K(x) where x ∈ F is transcendental over K. Let E be a finite separable extension of F . Let N(E) and g(E) denote the number of places of degree one (or rational places), and the genus, respectively, of E. Let [E : F ] denote the degree of this extension. In recen...