ar X iv : m at h / 06 05 45 4 v 1 [ m at h . M G ] 1 6 M ay 2 00 6 Ahlfors - Regular Curves In Metric Spaces

ar X iv : h ep - t h / 06 05 27 4 v 1 2 8 M ay 2 00 6 hep - th / 0605274 Metric and coupling reversal in string theory

Invariance under reversing the sign of the metric G M N (x) and/or the sign of the string coupling field H Universe K contains only F1 and NS5 of IIB and Heterotic SO(32).

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 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.