8 December ~ Christos Sarakasidis “Castelnuovo-Mumford regularity and the Eisenbud-Goto conjecture.”

The Castelnuovo-Mumford (CM) regularity is considered as one of the main invariants in (projective) algebraic geometry and commutative algebra. It gives information of the chosen embedding of a projective scheme inside a projective space. It’s been introduced by Castelnuovo and later on Mumford used the same definition in terms of sheaves to solve moduli problems. Since then, CM regularity has been found to be extremely useful for many reasons in combinatorics and module theory too. A famous conjecture by Eisenbud-Goto made on the upper bound for the CM regularity over arbitrary field for non-degenerate projective schemes in 1982. In 2010, Irena Peeva gave a counter-example for projective schemes in any dimension and arbitrary field. Though if we restrict to certain classes of geometrical objects (smooth schemes mostly) we obtain this nice upper bound given by the Eisenbud-Goto conjecture. During this talk we shall go through some of the above facts and delineate some of the progress around this conjecture.