ar X iv : m at h / 06 05 37 9 v 1 [ m at h . R A ] 1 5 M ay 2 00 6 AN EXTENDED FREUDENTHAL MAGIC SQUARE IN CHARACTERISTIC 3

ar X iv : m at h / 06 05 35 5 v 1 [ m at h . G R ] 1 3 M ay 2 00 6 Axes in Outer Space

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.

ar X iv : m at h / 06 05 36 9 v 1 [ m at h . A G ] 1 4 M ay 2 00 6 Vanishing Cycles and Thom ’ s a f Condition ∗

We give a complete description of the relationship between the vanishing cycles of a complex of sheaves along a function f and Thom’s af condition.

ar X iv : m at h / 06 05 44 4 v 1 [ m at h . N T ] 1 6 M ay 2 00 6 ABELIAN VARIETIES OVER CYCLIC FIELDS

Let K be a field of characteristic 6= 2 such that every finite separable extension of K is cyclic. Let A be an abelian variety over K. If K is infinite, then A(K) is Zariski-dense in A. Unless K ⊂ F̄p for some p, the rank of A over K is infinite.

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.