Birational Models of Singular Fano 3-folds

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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.