2 6 M ay 2 00 6 Some Remarks on the Local Fundamental Group Scheme

2 5 M ay 2 00 6 Some Remarks on the Local Fundamental Group Scheme

ar X iv : m at h / 06 05 64 5 v 1 [ m at h . A G ] 2 4 M ay 2 00 6 ON NORI ’ S FUNDAMENTAL GROUP SCHEME

The aim of this note is to give two structure theorems on Nori’s fundamental group scheme of a proper connected variety defined over a perfect field and endowed with a rational point.

1 6 M ay 2 00 6 Remarks on zeta functions and K - theory over F 1

Text of Abstract

2 2 M ay 2 00 6 Remarks on zeta functions and K - theory over F 1

We show that the notion of zeta functions over the field of one element F1, as given in special cases by Soulé, extends naturally to all F1-schemes as defined by the author in an earlier paper. We further give two constructions of K-theory for affine schemes or F1-rings, we show that these coincide in the group case, but not in general.

ar X iv : m at h / 06 05 37 3 v 1 [ m at h . A G ] 1 5 M ay 2 00 6 FLATNESS OF THE LINKED GRASSMANNIAN

We show that the linked Grassmannian scheme, which arises in a functorial compactification of spaces of limit linear series, and in local models of certain Shimura varieties, is Cohen-Macaulay, reduced, and flat. We give an application to spaces of limit linear series.