On Lyubeznik numbers of projective schemes

TL;DR

This paper investigates whether Lyubeznik numbers of projective schemes depend solely on the scheme itself or also on the embedding, showing that in positive characteristic, these numbers take only finitely many values across all embeddings.

Contribution

It proves that in positive characteristic, Lyubeznik numbers for a fixed projective scheme have finitely many possible values regardless of embedding.

Findings

Lyubeznik numbers depend only on the scheme in positive characteristic.

Finitely many Lyubeznik number values occur across all embeddings for a fixed scheme.

Progress toward embedding independence of Lyubeznik numbers in positive characteristic.

Abstract

Let $X$ be an arbitrary projective scheme over a field $k$. Let $A$ be the local ring at the vertex of the affine cone for some embedding $\iota: X\hookrightarrow \mathbb{P}^n_k$. G. Lyubeznik asked (in \cite{l2}) whether the integers $\lambda_{i,j}(A)$ (defined in \cite{l1}), called the Lyubeznik numbers of $A$, depend only on $X$, but not on the embedding. In this paper, we make a big step toward a positive answer to this question by proving that in positive characteristic, for a fixed $X$, the Lyubezink numbers $\lambda_{i,j}(A)$ of the local ring $A$, can only achieve finitely many possible values under all choices of embeddings.