EPSRC Reference: 
EP/T015896/1 
Title: 
Birational Models of Singular Fano 3folds 
Principal Investigator: 
Abban, Dr H 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematical Sciences 
Organisation: 
Loughborough University 
Scheme: 
New Investigator Award 
Starts: 
01 October 2020 
Ends: 
30 September 2022 
Value (£): 
234,599

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
The birational classification of Fano varieties in dimension 3, socalled 3folds, 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 3folds. 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 3folds without singularities have been studied extensively. A singularity is a point on a Fano 3fold at which the concept of tangency fails to make sense, like the sharp edge of an icecream cone. The Minimal Model Program, the main tool in birational geometry, indicates that Fano 3folds may carry mild singularities, the socalled terminal singularities. Hence the study of singular models is vital. We know that there are at most 52,000 families of Fano 3folds, from which we can construct only a few hundred, but most others remain mysteriously unconstructed. This obstruction may be resolved: most unconstructed Fano 3folds 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. Nonsolid Fano 3folds are hence less interesting, as the pencil model has more geometric information to offer. The algebraic structure of the unconstructed Fano 3folds 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 3folds in order to develop a better understanding of this mysterious corner of mathematics.

Key Findings 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Potential use in nonacademic contexts 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Impacts 
Description 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk 
Summary 

Date Materialised 


Sectors submitted by the Researcher 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Project URL: 

Further Information: 

Organisation Website: 
http://www.lboro.ac.uk 