  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 one. More generally, these
invariants determine a bound on connectedness dimension. Additionally, our
methods imply that the Lyubeznik number with indices (1,2) of the local ring at
the vertex of the affine cone over a projective variety is independent of the
choice of its embedding into projective space.
