Positivity properties of toric line bundles and tropical divisors
Lead Research Organisation:
University of Edinburgh
Department Name: Sch of Mathematics
Abstract
An algebraic variety is a subset of space that is given as the set of points where finitely many polynomial equations vanish. Intrinsic in this definition is an interplay between geometric properties of the variety and algebraic properties of the defining equations, a classical and fundamental problem in algebraic geometry. The projects described in this proposal will provide a major step towards the understanding of this relationship.
Hilbert realised that a powerful way of studying the defining equations of a variety is by considering the so-called higher syzygies. A first syzygy is a polynomial relations satisfied by the given equations. There are finitely many first syzygies that generate (allowing polynomial coefficients) all first syzygies of a given set of equations. Now one can continue by looking at the relations between the first syzygies, the so-called second syzygies, and so on. The Betti numbers record the number of ith syzygies of a certain degree needed to generate all syzygies. They carry a lot of information about the embedding of the variety. For example, given the equations vanishing on seven points in space, one can detect from their Betti numbers whether there exists a polynomial of degree 3 vanishing on all of them.
Usually one studies algebraic varieties from the abstract point of view, admitting many different embeddings into space that correspond to so-called very ample divisors, certain formal sums of subvarieties of one less dimension. It is a fundamental question in algebraic geometry to study the relationship between geometric properties of these divisors and algebraic properties of the resulting embedding. An example of a geometric property would be in how many points the divisor intersects an arbitrary given curve, and an example of an algebraic property would be that all defining equations can be generated from equations of degree two. This proposal deals with investigating this relationship using methods of discrete geometry.
Hilbert realised that a powerful way of studying the defining equations of a variety is by considering the so-called higher syzygies. A first syzygy is a polynomial relations satisfied by the given equations. There are finitely many first syzygies that generate (allowing polynomial coefficients) all first syzygies of a given set of equations. Now one can continue by looking at the relations between the first syzygies, the so-called second syzygies, and so on. The Betti numbers record the number of ith syzygies of a certain degree needed to generate all syzygies. They carry a lot of information about the embedding of the variety. For example, given the equations vanishing on seven points in space, one can detect from their Betti numbers whether there exists a polynomial of degree 3 vanishing on all of them.
Usually one studies algebraic varieties from the abstract point of view, admitting many different embeddings into space that correspond to so-called very ample divisors, certain formal sums of subvarieties of one less dimension. It is a fundamental question in algebraic geometry to study the relationship between geometric properties of these divisors and algebraic properties of the resulting embedding. An example of a geometric property would be in how many points the divisor intersects an arbitrary given curve, and an example of an algebraic property would be that all defining equations can be generated from equations of degree two. This proposal deals with investigating this relationship using methods of discrete geometry.
Planned Impact
The primary impact of this project will be on further developing expertise in the UK in the field of algebraic geometry, tropical geometry, commutative algebra, and combinatorics. This will maintain and increase the world-leading role of UK researchers in these areas, which has been recognised in the past two international reviews of mathematics.
This project addresses important unsolved problems about the structure of the defining ideals of algebraic varieties. It will use a completely new combinatorial approach to the study of such ideals for a special class of algebraic varieties, but will also have implications to ideals of more general algebraic varieties. This project will also have impact on researchers who need to find the defining equations of given ideals, in particular, in applied algebraic geometry.
A key part of the proposed grant will be the training of a postdoctoral researcher (PDRA). Thus this project will have a direct impact on building the capacity of the UK in the mathematical sciences and in STEM fields more generally. Research in pure mathematics is of benefit to society more broadly. More generally, mathematics is fundamental to the UK economy as explained in a recent study by EPSRC. As well as these direct impacts, mathematical research activities by organisations and employees also has indirect and induced effects.
This project addresses important unsolved problems about the structure of the defining ideals of algebraic varieties. It will use a completely new combinatorial approach to the study of such ideals for a special class of algebraic varieties, but will also have implications to ideals of more general algebraic varieties. This project will also have impact on researchers who need to find the defining equations of given ideals, in particular, in applied algebraic geometry.
A key part of the proposed grant will be the training of a postdoctoral researcher (PDRA). Thus this project will have a direct impact on building the capacity of the UK in the mathematical sciences and in STEM fields more generally. Research in pure mathematics is of benefit to society more broadly. More generally, mathematics is fundamental to the UK economy as explained in a recent study by EPSRC. As well as these direct impacts, mathematical research activities by organisations and employees also has indirect and induced effects.
People |
ORCID iD |
| Milena Hering (Principal Investigator) |
Publications
Ardila F
(2015)
The closure of a linear space in a product of lines
in Journal of Algebraic Combinatorics
Boocher A
(2016)
On the growth of deviations
in Proceedings of the American Mathematical Society
Boocher A
(2015)
Robust Graph Ideals
in Annals of Combinatorics
Boocher Adam
(2015)
EDGE IDEALS AND DG ALGEBRA RESOLUTIONS
in MATEMATICHE
Bossinger L
(2018)
Toric Degenerations of Gr(2, n) and Gr(3, 6) via Plabic Graphs
in Annals of Combinatorics
Chou J
(2016)
Diagonal splittings of toric varieties and unimodularity
Chou J
(2017)
Diagonal splittings of toric varieties and unimodularity
in Proceedings of the American Mathematical Society
| Description | While the defining equations of Veronese embeddings of projective spaces are well understood, the Betti numbers, i.e., the numbers of higher relations (syzygies) between these are still a mystery, even for embeddings of the projective plane. One key towards understanding these is by computing the cohomology of so-called Lazarsfeld-Mukai bundles, vector bundles that are associated to the given embedding. With Adam Boocher, the postdoc I hired on the grant, we explored one possible avenue of computing this cohomology, via a description of the cohomology due to Klyachko. Using this we are able to give a way to computing Betti numbers in terms of intersections of certain vector spaces. Unfortunately, we were not able to understand these intersections very well yet, but we are preparing a publication that will be accessible to researchers in matroid theory and combinatorics, in the hope that they have the necessary tools to solve the resulting problems. During the period of the grant I have established a new collaboration with Wouter Castryck and Alex Lemnens. The Betti numbers of toric embeddings are supported on certain sets of lattice points,. While we are not obtaining a complete understanding of these sets, we were able to show that they are much smaller than previously known. Computing defining equations of toric varieties is a ubiquitous problem in many applications, and our results mean that a computer computing defining equations and syzygies of toric varieties has to check fewer cases. During the period of the grant, I have established a research collaboration with Hendrik Süss where we study stability conditions for toric vector bundles. We were able to give a nice combinatorial criterion for the stability of the tangent bundle with respect to a given embedding in terms of the polytope corresponding to the embedding. In work with Süss and Nill, we use this criterion to get a very good understanding of the case of toric surfaces, a conjectural understanding of rational surfaces, as well as a decent understanding of toric varieties of Picard rank 2. Our methods open up a the possibility to improve our understanding of the relationship between stability to tangent bundles and the classification of algebraic varieties (the minimal model program), at least in the toric case. We were also able to give a combinatorial criterion for Lazarsfeld-Mukai bundles to be stable, which opens up new research questions when this combinatorial condition is satisfied. Once this is completed, one can apply the machinery of stability conditions to compute he cohomology of these bundles. In a project with Kevin Tucker, we study semi-normal toric varieties and how the Frobenius morphism acts on them. This allows us to compute many examples of great interest to commutative algebra such as the F-splitting ratio (a measure of the singularities of the ring), the test ideal and related ideals, and the decomposition of the push-forward of a toric semigroup ring by the Frobenius morphism into modules over itself. |
| Exploitation Route | Further research in algebraic geometry, commutative algebra, and algebraic statistics. Applications to algebraic statistics have the chance to be used in many other fields, such as biology. |
| Sectors | Education Other |
| URL | https://www.maths.ed.ac.uk/~mhering/ |
| Description | The article on phase retrieval has had an impact in the academic community of frame theory and its applications, mainly signal processing. The other articles have had impact within the academic communities of algebraic geometry and commutative algebra. |
| Description | Adam in Italy |
| Organisation | University of Catania |
| Country | Italy |
| Sector | Academic/University |
| PI Contribution | Adam Boocher went to participate in the 2015 program Pragmatic at the University of Catania. |
| Collaborator Contribution | Catania hosted the conference. |
| Impact | Ardila F, Boocher A. (2015). The closure of a linear space in a product of lines. Journal of Algebraic Combinatorics, (1), doi: 10.1007/s10801-015-0634-x Boocher A, Brown B, Duff T, Lyman L, Murayama T, Nesky A, ... Schaefer K. (2015). Robust Graph Ideals. Annals of Combinatorics, (4), doi: 10.1007/s00026-015-0288-3 Boocher A, D'Alì A, Grifo E, Montaño J, Sammartano A. (2016). On the growth of deviations. Proceedings of the American Mathematical Society, (12), doi: 10.1090/proc/13132 Boocher Adam, D'Ali Alessio, Grifo Eloisa, Montano Jonathan, Sammartano Alessio. (2015). EDGE IDEALS AND DG ALGEBRA RESOLUTIONS. MATEMATICHE, 70(1), pp. 215-238. doi: 10.4418/2015.70.1.16 |
| Start Year | 2015 |
| Description | CastryckLemmens |
| Organisation | University of Leuven |
| Country | Belgium |
| Sector | Academic/University |
| PI Contribution | We are preparing an article, and our contributions were equal. |
| Collaborator Contribution | We are preparing an article, and our contributions were equal. |
| Impact | We are preparing an article described in the section on research outcomes. |
| Start Year | 2015 |
| Description | Collaboration with Adam Boocher |
| Organisation | University of Edinburgh |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | Adam Boocher is the postdoc I hired. We have been establishing a research relationship and are in the process of completing an article. |
| Collaborator Contribution | We participated equally in the research for our article. Adam is very good at programming, and he did the programming part of the collaboration. |
| Impact | An article in preparation on computing the cohomology of Lazarsfeld-Mukai bundles on toric varieties. |
| Start Year | 2014 |
| Description | TropicalGrassmannian |
| Organisation | University of Edinburgh |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | We wrote an article, the contribution was equal |
| Collaborator Contribution | We wrote an article, the contribution was equal |
| Impact | Bossinger L, Fang X, Fourier G, Hering M, Lanini M. (2018). Toric Degenerations of Gr(2, n) and Gr(3, 6) via Plabic Graphs. Annals of Combinatorics, (3), doi: 10.1007/s00026-018-0395-z |
| Start Year | 2016 |
| Description | ICMS Betti workshop |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Postgraduate students |
| Results and Impact | With Frank-Olaf Schreyer I co-organised a conference at ICMS in 2015 on Minimal Free Resolutions, Betti numbers, and Combinatorics. The conference was a big success, it attracted more than 70 participants and inspired new research. |
| Year(s) Of Engagement Activity | 2015 |