Connectedness and Lyubeznik numbers

TL;DR

This paper explores how Lyubeznik numbers, invariants from local cohomology, relate to the connectedness properties of spectra, providing characterizations and bounds for connectedness dimension in algebraic geometry.

Contribution

It establishes that Lyubeznik numbers determine the connectedness dimension and shows their independence from embeddings for certain affine cones.

Findings

Lyubeznik numbers characterize connectedness dimension equals one.

They provide bounds on connectedness dimension.

The (1,2) Lyubeznik number is embedding-independent for affine cones over projective varieties.

Abstract

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.