The first main result of this paper is a bijective correspondence between the strictly full triangulated subcategories dense in a given triangulated category and the subgroups of its Grothendieck group (Thm. 2.1). Since every strictly full triangulated subcategory is dense in a uniquely determined thick triangulated subcategory, this result refines any known classification of thick subcategories to a classification of all strictly full triangulated ones. For example, one can thus refine the famous classification of the thick subcategories of the finite stable homotopy category given by the work of Devinatz–Hopkins–Smith ([Ho], [DHS], [HS] Thm. 7, [Ra] 3.4.3), which is responsible for most of the recent advances in stable homotopy theory. One can likewise refine the analogous classification given by Hopkins and Neeman ([Ho] Sect. 4, [Ne] 1.5) of the thick subcategories of $D(R)_{\rm parf}$, the chain homotopy category of bounded complexes of finitely generated projective $R$-modules, where $R$ is a commutative noetherian ring.

Let ${E}$ be a nonconstant elliptic curve, over a global field ${K}$ of positive, odd characterisitc. Assuming the finiteness of the Shafarevich-Tate group of ${E}$, we show that the order of the Shafarevich-Tate group of ${E}$, is given by${O}({N}^{1/2+6\,\log(2)/\log({q})})$, where ${N}$ is the conductor of ${E}, {q}$ is the cardinality of the finite field of constants of ${K}$, and where the constant in the bound depends only on ${K}$. The method of proof is to work with the geometric analog of the Birch-Swinnerton Dyer conjecture for the corresponding elliptic surface over the finite field, as formulated by Artin-Tate, and to examine the geometry of this elliptic surface.

We extend a result of Vojta on height inequalities for algebraic points on curves over function fields to include the case of positive characteristic. The main tool used is the Kodaira—Spencer map and destabilizing flags for vector bundles on curves.

A Pfaffian equation on $X$ is an invertible subsheaf $L \hookrightarrow \Omega^1_X$ of the sheaf of differential 1-forms on $X$. It can thus be thought of as a global section of $\Omega^1_X \otimes L^{-1}$ or, by choosing an isomorphism $L^{-1} \simeq {\cal O}_X(E) \subset K$, as a meromorphic differential form $\omega$ on $X$. We say that the Pfaffian equation has a first integral if there is a non-empty open subset $U\subset X$ and a smooth map $f: U\rightarrow {\rm P}^1$ such that $L_U \simeq f^* \Omega^1_{{\rm P}^1}$ as subsheaves of $\Omega^1_X$; that is, if $L$ ‘comes from’ a rational map to a curve. In this case, $f$, viewed as a rational function, is called a first integral. Note that as rational differential forms, ${\rm d} f \wedge \omega = 0$.

A zero set of a holomorphic vector field is totally degenerate, if the endomorphism of the conormal sheaf induced by the vector field is identically zero. By studying a class of foliations generalizing foliations of ${\rm C}^*$-actions, we show that if a projective manifold admits a holomorphic vector field with a smooth totally degenerate zero component, then the manifold is stably birational to that component of the zero set. When the vector field has an isolated totally degenerate zero, we prove that the manifold is rational. This is a special case of Carrell's conjecture.

A strategy is proposed for applying Chabauty's Theorem to hyperelliptic curves of genus ${>} 1$. In the genus 2 case, it shown how recent developments on the formal group of the Jacobian can be used to give a flexible and computationally viable method for applying this strategy. The details are described for a general curve of genus 2, and are then applied to find ${\bm C}({\bb Q})$ for a selection of curves. A fringe benefit is a more explicit proof of a result of Coleman.

We introduce some blow-analytic invariants of real analytic function-germs and discuss their properties. As a consequence, we obtain, for instance, the multiplicity of function-germs is a blow-analytic invariant.

In this paper we prove that the moduli spaces $MI_{\scriptstyle {\bb P}^{2n+1}}(k)$ of mathematical instanton bundles on ${\bb P}^{2n+1}_{{\bb C}}$ with quantum number $k$ are singular for $n \geqslant 2$ and $k \geqslant 3$, giving a positive answer to a conjecture made by Ancona and Ottaviani in 1993.