Classifying algebraic varieties via Newton-Okounkov bodies

Algebraic geometry is a branch of mathematics that studies geometric objects using tools from abstract algebra. The main characters, called algebraic varieties, are geometric shapes in the space (such as curves, surfaces) that can be described as the solution sets of collections of polynomial equations in several variables. We can also go a step further and attach a vector space to any point of an algebraic variety, obtaining what we call a vector bundle (a "bundle of vector spaces"), and then look at the properties of such an object as a whole.

In Mathematics we are interested in classification problems, that are solved by first choosing an equivalence relation, "two objects are equivalent if they satisfy a certain property", and then by listing all possible equivalence classes, sets of equivalent objects. In Geometry, for instance we say that two objects are "isomorphic" (from Ancient greek) if they have the same ("isos") shape ("morhpe"). The set whose elements are the equivalence classes with respect to isomorphism of a certain type of algebraic varieties or vector bundles forms what we call a moduli space.

Fano varieties are the algebraic varieties that have the simplest shape: they are "positively curves" and can be thought as being the higher dimensional version of the two dimensional sphere. Fano varieties provide a source of very explicit examples of algebraic varieties in general. Moreover they often occur in application to other subjects such as string theory.

This project is concerned with moduli spaces of curves and of certain vector bundles over them on the one hand, and with Fano varieties on the other hand. These are among the most beautiful and well-studied objects in algebraic geometry. Understanding the geometric behaviour of moduli spaces and classifying Fano varieties have been in the cutting-edge of algebraic geometric research, and are particularly important due to their connections with physics: with theoretical physics and string theory respectively.

The goals of this project is to show that it is possible to deform moduli spaces and Fano varieties into some easier objects, called toric varieties, and then to classify the former based on the properties of the latter. This procedure is called toric degeneration. Toric varieties are well understood algebro-geometric objects that can be described in terms of the combinatorial data encoded into convex polytopes in Euclidean space, the high dimensional version of convex polygons.

In general it is a very difficult problem to decide whether toric degenerations exist and, if so, how to obtain them in practice. The main tool that will be used in the project is the construction of Newton-Okounkov bodies. These are convex bodies in Euclidean space, named after I. Newton, as it generalised the Newton polygons, and A. Okounkov who in the 1990s brought this idea into the algebro-geometric setting. When these bodies have a nice combinatorial shape such as that of a polytope, we can construct toric degenerations of the corresponding algebraic variety and proceed with the classification.

The proposed research is in pure mathematics. Research in fundamental sciences and in particular in Mathematics may take a very long time to have a socio/economic impact. This makes it very difficult, if not even impossible, to predict if and how results obtained will have an impact and how.

While we cannot claim an immediate impact on society and economy, during the lifetime of this project we can surely aim at creating an impact on knowledge, by disseminating our scientific advances, and on people, by contributing through training of young researchers. With the scope of maximising the impact of this research, the PI will adopt the following dissemination strategy in order to reach an audience that is as broad as possible.

One-to-one discussions with visiting leading experts as well as during research visits to other institutes are vital for disseminating own research and influencing others' research, so to ensure that impact is made on the research of one another. The PI has strong links to senior researchers in the UK algebro-geometric community and, in addition, she has close contacts with research groups overseas in Italy (Florence, Milan, Padua, Rome, Trento, Turin), in Belgium (Leuven, Gent), in Warsaw (Cracow, Warsaw), in Scandinavia (Copenhagen, Stockholm, Oslo and Bergen), in France (Nice, Paris), in Germany (Bonn, Hannover, Frankfurt), in North America (Michigan, Texas, Illinois) and in South America (Buenos Aires, Concepcion, Rio de Janeiro) and in Asia (Seoul and Daejeon). These links will allow the research to be widely disseminated by means of seminars and workshops in the UK and abroad.

The PI will also disseminate her work by giving talks at international conferences in algebraic geometry, that will take place in the next years. As well as in the algebro-geometric community, this research will be disseminated to researchers of other fields of pure and applied Mathematics such as, for instance, mathematical physics and geometric modelling, in particular spline theory and CAD.