A norm for the cohomology of 2-complexes

We introduce a norm on the real 1-cohomology of finite 2-complexes determined by the Euler characteristics of graphs on these complexes. We also introduce twisted Alexander-Fox polynomials of groups and show that they give rise to norms on the real 1-cohomology of groups. Our main theorem states that for a finite 2-complex X, the norm on H^1(X; R) determined by graphs on X majorates the Alexander-Fox norms derived from \pi_1(X).Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-7.abs.htm

Components of Gr\"obner strata in the Hilbert scheme of points

We fix the lexicographic order $\prec$ on the polynomial ring $S=k[x_{1},...,x_{n}]$ over a ring $k$. We define \Hi^{\prec\Delta}_{S/k}, the moduli space of reduced Gr\"obner bases with a given finite standard set $\Delta$, and its open subscheme \Hi^{\prec\Delta,\et}_{S/k}, the moduli space of families of #\Delta points whose attached ideal has the standard set $\Delta$. We determine the number of irreducible and connected components of the latter scheme; we show that it is equidimensional over ${\rm Spec}\,k$; and we determine its relative dimension over ${\rm Spec} k$. We show that analogous statements do not hold for the scheme \Hi^{\prec\Delta}_{S/k}. Our results prove a version of a conjecture by Bernd Sturmfels.Comment: 49 page