Birational Models of Singular Fano 3-folds

The birational classification of Fano varieties in dimension 3, so-called 3-folds, has been a challenging problem in mathematics for decades. Two varieties are called birational if they can be identified after removing some small (algebraic) subsets from them. This project aims to shed light on the birational geometry of singular Fano 3-folds. Fano varieties, the objects of study here, are fundamental geometric shapes described as the solution sets of algebraic equations (polynomials) so that their geometry has some special positivity properties. Roughly speaking, they are positively curved. They appear in applications: for example, any geometric shape that can be parametrized by rational functions is approximated by Fano varieties.

Fano 3-folds without singularities have been studied extensively. A singularity is a point on a Fano 3-fold at which the concept of tangency fails to make sense, like the sharp edge of an ice-cream cone. The Minimal Model Program, the main tool in birational geometry, indicates that Fano 3-folds may carry mild singularities, the so-called terminal singularities. Hence the study of singular models is vital. We know that there are at most 52,000 families of Fano 3-folds, from which we can construct only a few hundred, but most others remain mysteriously unconstructed. This obstruction may be resolved: most unconstructed Fano 3-folds are most likely not solid. A Fano variety is called solid if it cannot be birational to a pencil, or web, of lower dimensional Fano varieties. Non-solid Fano 3-folds are hence less interesting, as the pencil model has more geometric information to offer. The algebraic structure of the unconstructed Fano 3-folds is similar to those that we know but with imposed terminal singularities: they all have complex pluri anticanonical rings. We will examine solidity for the singular Fano 3-folds in order to develop a better understanding of this mysterious corner of mathematics.

The core of this proposal is blue sky research in mathematics. Its instant implications fall within algebraic geometry, then in nearby subjects such as Arithmetic Geometry and Complex Geometry. Further impact on society and the economy are likely but will be carried out via other sciences and academic users. The most likely application will be in industrial design, through effective algorithms in geometric parametrisation. The research process itself will have tangible socio-economic impact through training of researchers and immediate influence on UK higher education competitiveness.
Knowledge: This project will make significant progress in the birational classification of Fano 3-folds with terminal singularities.
Society and Economy: The primary objective of this programme is to study birational relations amongst certain singular Fano 3-folds. These are representative models for a variety (a geometric object) that is potentially parametrisable. There are working algorithms, implemented in the computer algebra system MAGMA, that produce a parametrisation for a given (rational) surface. An effective such algorithm in dimension 3 is out of reach at the moment and is beyond the scope of this 2-year project. A major goal of this programme is to make significant steps forward in simplifying these representative models. In real-world examples, such 3-dimensional objects can illustrate a deforming parametrisable surface or, less geometrically, a model of big data that is represented as a 3-fold.
People: Given the central role that birational geometry plays in modern mathematics, having a dynamic and productive group will directly have a positive impact on UK higher education competitiveness. To add to this, I will hire a top young researcher as PDRA, either retaining a top mathematician in, or bringing new talent to, the UK. The PDRA will be trained in highly technical advanced mathematics, ready to serve the society as an academic or a competent skilled worker in the non-academic sector.