Elliptic three-folds II: Multiple fibres
Let f: X -* S be an elliptic fibration with a section, where S is a projective surface and X is a projective threefold. We determine when it is possible to perform a logarithmic transformation along a closed subset Z C S to obtain a new elliptic fibration f': X' -* S which now has multiple fibres along Z. This is done in the setting of Ogg-Shafarevich theory. We find a number of obstructions to performing such a logarithmic transformation, the very last of which takes values in the torsion part of the codimension 2 Chow group of X. 0.  Suppose that we would like to classify algebraic elliptic three-folds, i.e. algebraic three-folds X with a fibration f: X -* S whose general fibre is an elliptic curve. In analogy with the classification of elliptic surfaces, there are three steps: (1) Classify elliptic fibrations with section. These can all be obtained birationally using Weierstrass models. (2) Classify elliptic fibrations f' : X -* S without multiple fibres which have a given jacobian fibration f: J -* S. This has already been carried out in ; the set of such elliptic fibrations is parametrized up to birational equivalence by the Tate-Shafarevich group 11Is(A), where A is the generic fibre of f. (3) Understand how to perform logarithmic transformations. In the case of complex elliptic surfaces, a logarithmic transformation is an operation for producing multiple fibres over points in the base curve. It is a purely local operation, which can then be patched to give a new complex elliptic surface, which may or may not be algebraic. However, it is well understood when the resulting surface will be algebraic. In the case of threefolds, this is a considerably more complicated task. Now one should expect to perform logarithmic transformations along curves in the base surface, and so when dealing with a global base, one cannot expect a purely local description for a logarithmic transformation. It is step 3 which this paper carries out. Instead of the local analytic description of logarithmic transformations given by Kodaira in  and extended to higher dimensions by N. Nakayama in , we take the algebraic approach of Ogg-Shafarevich theory. , and also .) In this point of view, Received by the editors June 19, 1995.  This material is based upon work supported by the North Atlantic Treaty Organization under a Grant awarded in 1990. Research at MSRI supported in part by NSF grant #DMS 9022140.

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: Elliptic Three-folds I: Ogg-Shafarevich Theory
abstract: 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: A finiteness theorem for elliptic Calabi-Yau threefolds
abstract: We prove that up to birational equivalence, there exists only a finite number of families of Calabi-Yau threefolds (i.e. a threefold with trivial canonical class and factorial terminal singularities) which have an elliptic fibration to a rational surface. This strengthens a result of B. Hunt that there are only a finite number of possible Euler characteristics for such threefolds.

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: 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.