Seminars

Tuesday October 11th in room Byron Blanc at 10:30am

Nicolas Botbol
INRIA - GALAAD

will speak

About Castelnuovo-Mumford regularity

In this talk I will overview some historical and classical results around Castelnuovo-Mumford regularity finishing with some of the latest developments in the area.
Castelnuovo-Mumford regularity is one of the most fundamental invariants in Commutative Algebra and Algebraic Geometry. This invariant was tacitly introduced by Castelnuovo in 1893 and formally defined, under the name of Castelnuovo regularity, by Mumford in 1966. Mumford's definition is in terms of sheaf cohomology, language introduced by Serre in his fundamental work FAC of 1955. In 1982, Ooishi, later complemented by Eisenbud and Goto, gave a "commutative algebra" version of Castelnuovo-Mumford regularity in terms of local cohomology and minimal free resolutions, making explicit the relation between cohomological aspects of projective varieties and commutative algebra, already noticed by Mumford and indeed suggested by Castelnuovo in his original work.
Years later, in 1995, Cox associates to an abstract toric variety X a polynomial ring R, graded by the divisor class group of X, which he called The Homogeneous Coordinate Ring of X. This has motivated during the last decade several multigraded versions of Castelnuovo-Mumford regularity, that propelled and continue propelling many results in the area.