Koszul cohomology, Syzygies and Geometry
Bangere Purnaprajna (Kansas)

The topic of Syzygies of projective varieties has attracted attention after the work of Mark Green in the early eighties. Green's work showed that the classical results on projective normality and normal presentation (by Castelnuovo and Mumford among others) were part of a more general phenomenon involving the so-called higher syzygies. One of the heavily pursued topic in this area is to determine the linearity (or the failure of linearity after a certain stage)of the resolution of the homogeneous coordinate ring associated to an embedding of a variety $X$ given by a very ample line bundle $L$. To relate the failure of the linearity (after a certain stage) to the geometry of the variety is one of the main goals. All of this involves computing some Koszul cohomology groups. This area is reasonably well charted for curves and not so for surfaces and higher dimensions. I will talk mostly regarding some recent results regarding adjunction bundles on surfaces of general type. It must be noted that these questions are not even well posed for an arbitrary variety of dimenson 3 or higher due to non-availability of very ampleness results for adjunction bundles associated to an ample line bundle. I will try to illustrate some general principles that hold involving failure of linearity, Koszul cohomology and its relation to geometry through an example.

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