Alexander Engström: Finding extremal objects in Algebraic Geometry using Graph Theory

In Graph Theory the enumeration and existence of certain subgraphs are central invariants, and in extremal graph theory their relations are studied. For a particular graph it is, in some sense, easy to calculate the number of subgraphs of a certain type, but to provide general useful inequalities between subgraph counts is very hard.

In Algebraic Geometry the opposite picture have crystallised during the last five years: The invariants (fine graded Betti numbers) are still very hard to compute for a particular module, but there is a beautiful explicit polyhedral cone giving the linear inequalities relating them. With these inequalities issues open since Hilbert’s days are now answered or understood much better.

These linear inequalities are not always sufficient to characterise the invariants, but for a basic class of modules they are. To prove that we [1] went to graph theory to construct graphs where the invariants are possible to control and then converted them into modules.

In this talk I will try to explain by examples, assuming no particular background, how Graph Theory could help out in Algebraic Geometry.

[1] Alexander Engström and Matthew T. Stamps. Betti diagrams from graphs, Algebra Number Theory 7 (2013), no. 7, 1725-1742.