abstract: A graph is called "R-connected" if between any 2 vertices one can find R vertex-disjoint paths. For example, the graph of a polygon is 2-connected. More generally, Balinski's theorem says that the graph of every d-dimensional polytope is d-connected.
Hartshorne's connectedness theorem says that arithmetically Cohen-Macaulay schemes are connected in codimension one. We show a quantitative version of this result: If X is an arithmetically Gorenstein subspace arrangement, then the dual graph of X is r-connected, where r is precisely the Castelnuovo-Mumford regularity. In the very special case when X is the Stanley-Reisner variety of the boundary of a polytope, this recovers Balinski's theorem.
If time permits, we also discuss a very recent extension to dual graphs of projective curves, which is work in progress with Matteo Varbaro and Barbara Bolognese (Northeastern).