ABSTRACT OF THE PROJECT:
This project addresses a fundamental question on the relation between combinatorics and geometry. Mathematicians can prove theorems in combinatorics using algebraic geometry. Vice versa, we can do algebraic geometry also in absence of the algebraic variety by using combinatorics. These leitmotifs ware recently discovered by the field medalist June Huh in a linear setting.
The question is: Huh’s philosophy depends on the linear setting? Does it apply in much broader generality? The project will answer to this question by studying the toric case. This case is the middle point between the linear setting and the generic case.
The project is multidisciplinary, sitting between combinatorics, geometry, algebra, and topology. PI’s knowledge in the field of arrangements and matroids is an almost unique and necessary mix to carry out this research.
The aim is to show that almost all results about matroids – obtained in the last five years using algebraic geometry – can be extended to arithmetic matroids. The research will also increase the knowledge of toric arrangements and make them a new toy example and a new local model for all geometric theories. Concretely, the research group will invent a machinery that for almost any poset produces an algebra, theorems for that algebra, and inequalities between the numerical invariants of the given poset.
Methods used in this research will be various, including sheaves on posets, Chern theory, toric and tropical geometry. The last part of the project consists in applications of the machinery to combinatorics and algebraic geometry; in particular to polytope theory, resolution of singularities and moduli spaces.
The main idea is to define a class of posets that behaves like the poset of flats of a toric arrangement; from that poset construct a Chow ring of the poset, prove the Hodge package, the relative Lefschetz decomposition, and describe the NEF cone. Then, the research group will define and study important classes in the Chow ring of the poset such as CSM classes and Chern classes of a tautological bundle.
Finally, the intersection numbers of these classes will be related to the combinatorics of the poset. The aforementioned theorems, inspired by the algebraic geometry, will provide new inequalities for such numerical invariants.