Regularity 3 in edge ideals associated to bipartite graphs

A classical theorem in Combinatorial Algebra due to R. Fröberg ([Frö90])
gives a characterization of edge ideals with linear resolution or, equivalently, having regularity
2. This result was rened in [EGHP05] by determining in a pure combinatorial
way the first step in the minimal free resolution where nonlinear syzygies appear if the
resolution is not linear, and in [FG09] by proving that those first nonlinear syzygies all
have the same degree. The corresponding multigraded Betti numbers are also determined
in terms of the combinatorics of the graph in the last reference mentioned.
In this talk we will go one step further in the case of edge ideals associated to bipartite
graphs giving analogue characterization and renements for ideals with regularity 3. This
is the greatest extent possible without taking into account the characteristic of the ground
field.

References

[EGHP05] David Eisenbud, Mark Green, Klaus Hulek, and Sorin Popescu. Restricting linear syzygies: algebra
and geometry. Compositio Mathematica, 141(6):14601478, 2005.

[FG09] Oscar Fernandez-Ramos and Philippe Gimenez. First nonlinear syzygies of ideals associated to
graphs. Communications in Algebra, 37(6):19211933, June 2009.

[Frö90] Ralf Fröberg. On Stanley-Reisner rings. In Topics in algebra, Part 2 (Warsaw, 1988), volume 26
of Banach Center Publ., pages 5770. PWN, Warsaw, 1990.