Lyubeznik numbers of projective schemes

TL;DR

This paper proves that Lyubeznik numbers of projective schemes over a field of positive characteristic are intrinsic invariants, independent of the embedding, thus revealing their fundamental geometric nature.

Contribution

It establishes that Lyubeznik numbers for projective schemes in positive characteristic depend solely on the scheme itself, not on the embedding, highlighting their intrinsic geometric significance.

Findings

Lyubeznik numbers are intrinsic for projective schemes over fields of positive characteristic

Dependence of Lyubeznik numbers on the scheme rather than the embedding

Provides a new understanding of the geometric nature of Lyubeznik numbers

Abstract

Let $X$ be a projective scheme over a field $k$ and let $A$ be the local ring at the vertex of the affine cone of $X$ under some embedding $X\hookrightarrow\mathbb{P}^n_k$. We prove that, when $\ch(k)>0$, the Lyubeznik numbers $\lambda_{i,j}(A)$ are intrinsic numerical invariants of $X$, i.e., $\lambda_{i,j}(A)$ depend only on $X$, but not on the embedding.