Connectedness and Lyubeznik Numbers

We investigate the relationship between connectedness properties of spectra and the Lyubeznik numbers, numerical invariants defined via local cohomology. We prove that for complete equidimensional local rings, the Lyubeznik numbers characterize when connectedness dimension equals 1. More generally, these invariants determine a bound on connectedness dimension. Additionally, our methods imply that the Lyubeznik number $\lambda _{1,2}(A)$ of the local ring $A$ at the vertex of the affine cone over a projective variety is independent of the choice of its embedding into projective space.