ar X iv : m at h / 05 01 05 1 v 2 [ m at h . G R ] 1 7 M ay 2 00 6 INJECTIONS OF ARTIN GROUPS

ar X iv : m at h / 05 01 05 1 v 1 [ m at h . G R ] 4 J an 2 00 5 INJECTIONS OF ARTIN GROUPS

We study those Artin groups which, modulo their centers, are finite index subgroups of the mapping class group of a punctured sphere. In particular, we show that any injective homomorphism between these groups is parameterized by a homeomorphism of a punctured disk together with a homomorphism to the integers. The technique, following Ivanov, is to prove that every superinjective map of the com...

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 / 05 02 57 3 v 6 [ m at h . A G ] 1 4 M ay 2 00 6 ON DEFORMATIONS OF FLAG

Any (global) deformation of a flag manifold F with b2 = 1 is biholomorphic to F .

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

Date: May 15, 2006. 2000 Mathematics Subject Classification. Primary 17B25.

ar X iv : m at h / 01 05 15 1 v 1 [ m at h . A C ] 1 7 M ay 2 00 1 The Hilbert Series of Pfaffian Rings ⋆

We give three determinantal expressions for the Hilbert series as well as the Hilbert function of a Pfaffian ring, and a closed form product formula for its multiplicity. An appendix outlining some basic facts about degeneracy loci and applications to multiplicity formulae for Pfaffian rings is also included.