On Lyubeznik numbers of projective schemes

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 $\iota :X\hookrightarrow {\mathbb{P}}_k^n$. 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.