In Math 631, you learn about the configuration X of 27 lines on a smooth cubic surface in projective 3-space, reducing to the Fermat cubic case; moreover, each line in said configuration meets exactly 10 other lines. There's a notion of dual graph for projective subschemes, the dual graph of X is 10-regular and 10-connected (graph theory notions), and X has Castelnuovo-Mumford regularity 11. As part of a larger narrative, Benedetti-Di Marca-Varbaro (https://arxiv.org/abs/1608.02134) deduce a theorem involving dual graphs of arithmetically-Gorenstein line configurations with planar singularities, which places this factoid from Math 631 into a general framework of enumerative geometry facts like it. They give seven examples in 3-space to illustrate their work, and hopefully I'll have time to mention Example F (F=Fermat surfaces). With a view towards stating Benedetti-Di Marca-Varbaro's main theorem, I'll first try to survey the notions of dual graph and Castelnuovo-Mumford regularity as they pertain to complete intersection projective subspace arrangements.

This talk won't have proofs. My goal is to survey a handful of results more or less chronologically, and have a few examples on hand for illustration. The curious audience member can check out the relevant papers for self-study, or stop into Student Commutative Algebra where we try to work through the more algebraic facets of their work. Speaker(s): Robert Walker (UM)