Elliptic Three-folds I: Ogg-Shafarevich Theory
We calculate the Tate-Shafarevich group of an elliptic three-fold $f:X\rightarrow S$ when $X$ and $S$ are regular and $f$ is flat, relating it to the Brauer group of $X$ and $S$. We show that given certain hypotheses on $f$, the Tate-Shafarevich group has the interpretation of isomorphism classes of elliptic curves over the function field of $S$ which have the same jacobian as the generic fibre of $f$, and for which there exists a relatively minimal model which has no multiple fibres. We use this to give examples of elliptic fibrations with isolated multiple fibres, and also to give a new counterexample to the Luroth problem in dimension three. This is a revised, hopefully improved, version with a few extra theorems and a few errors corrected.

title: The distribution of bidegrees of smooth surfaces in Gr(1, P3)
abstract: The study of algebraic surfaces in Gr (1, p3), the Grassmann variety parametrising lines in projective three-space, was a popular one for algebraic geometers of the late nineteenth and early twentieth centuries. Calling them line congruences, researchers such as Kummer, Fano, Roth and many others published many papers on the topic, classifying congruences and studying their invariants. Since that time, the field has lain dormant until very recently. The classical geometers identified two numbers associated with a given congruence: the order and the class. Thinking of a congruence as a two dimensional family of lines, the order is the number of lines in the family passing through a general point in p3, and the class the number of lines in the family contained in a general plane. Together, these two numbers make up the bidegree of the congruence. In modern terms, the bidegree gives the class of the congruence in the Chow ring of Gr (1, p3). In this paper, we consider the question: "for what values of a and b does there exist (or not exist) a smooth congruence of bidegree (a, b)?" In particular, we try to find restrictions on the bidegree, using an approach suggested by Dolgachev and Reider in . This approach is to study the restriction of the universal bundle g of Gr (1, p3), which appears in the exact sequence

title: Deforming Calabi-Yau threefolds

abstract: This paper first generalises the Bogomolov-Tian-Todorov unobstructedness theorem to the case of Calabi-Yau threefolds with canonical singularities. The deformation space of such a Calabi-Yau threefold is no longer smooth, but the general principle is that the obstructions to deforming such a threefold are precisely the obstructions to deforming the singularities of the threefold. Secondly, these results are applied to smoothing singular Calabi-Yau threefolds with crepant resolutions. Any such Calabi-Yau threefold with isolated complete intersection singularities which are not ordinary double points is smoothable. A Calabi-Yau threefold with non-complete intersection isolated singularities is proved to be smoothable under much stronger hypotheses.

title: Equations of 
$(1,d)$-polarized abelian surfaces
abstract: In this paper, we study the equations of projectively embedded abelian surfaces with a polarization of type (1, d ). Classical results say that given an ample line bundle L on an abelian surface A, the line bundle L ⊗n is very ample for n ≥ 3, and furthermore, in case n is even and n ≥ 4, the generators of the homogeneous ideal IA of the embedding of A via L ⊗n are all quadratic; a possible choice for a set of generators of IA are the Riemann theta relations. On the other hand, much less is known about embeddings via line bundles L of type (1, d ), that is line bundles L which are not powers of another line bundle on A. It is well-known that if d ≥ 5, and A is a general abelian surface, then L is very ample, while L can never be very ample for d < 5. However, even if d ≥ 5, L may not be very ample for special abelian surfaces. We will restrict our attention in what follows only to the general abelian surface and wish to know what form the equations take for such a projectively embedded abelian surface. A few special cases are well-documented in the literature: d = 4, in which case the general surface is a singular octic in P3, cf. [BLvS], and d = 5 in which case the abelian surface is described as the zero set of a section of the Horrocks-Mumford bundle [HM], whereas its homogeneous ideal is generated by 3 (Heisenberg invariant) quintics and 15 sextics (cf. . Also, recent work by Manolache and Schreyer [MS] and by Ranestad [Ra] provides a description of the equations and syzygies in the case d = 7.

title: Special Lagrangian Fibrations I: Topology
abstract: In 1996, Strominger, Yau and Zaslow made a conjecture about the geometric relationship between two mirror Calabi-Yau manifolds. Roughly put, if X and Y are a mirror pair of such manifolds, then X should possess a special Lagrangian torus fibration $f:X\to B$ such that Y is obtained by dualizing the fibration f. This leaves a huge amount to be done to verify such conjectures. This paper takes a speculative point of view, in that it assumes that a special Lagrangian torus fibration exists on X. We address a number of questions of a topological nature: what is the relationship between the cohomology of X and the cohomology of the dual fibration? what kind of information does the Leray spectral sequence for f contain? what is the relationship between the topological (1,1) couplings of the dual of f and the (1,n-1)-couplings of X in the large complex structure limit? These questions are shown to have nice answers if a key conjecture about the monodromy diffeomorphisms about a large complex structure limit point holds. Roughly put, this conjecture says that monodromy about a large complex structure limit point can be described as a very natural generalization of a Dehn twist for an elliptic curve. Given this conjecture, we show, among other results, that the large complex radius limit of the (1,n-1) couplings on X coincide with the topological (1,1) couplings on Y, and if dim X=3, the Leray filtration and weight filtrations of the mixed Hodge structure coincide, as conjectured by myself and P.M.H. Wilson, and independently by D. Morrison.

title: Primitive Calabi-Yau threefolds
abstract: A primitive Calabi-Yau threefold is a non-singular Calabi-Yau threefold which cannot be written as a crepant resolution of a singular fibre of a degeneration of Calabi-Yau threefolds. These should be thought as the most basic Calabi-Yau manifolds; all others should arise through degenerations of these. This paper first continues the study of smoothability of Calabi-Yau threefolds with canonical singularities begun in the author's previous paper, ``Deforming Calabi-Yau Threefolds.'' (alg-geom 9506022). If X' is a non-singular Calabi-Yau threefold, and f:X'->X is a contraction of a divisor to a curve, we obtain results on when X is smoothable. We then discuss applications of this result to the classification of primitive Calabi-Yau threefolds. Combining the deformation theoretic results of this paper with those of ``Deforming Calabi-Yau Threefolds,'' we obtain strong restrictions on the class of primitive Calabi-Yau threefolds. We also speculate on how such work might yield results on connecting together all moduli spaces of Calabi-Yau threefolds.

title: Tensor models and simplicial quantum gravity in >2-D
abstract: Various tensor model formulations of simplicial quantum gravity are presented and discussed but not solved.