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Let $X$ be an arbitrary projective scheme over a field $k$. Let $A$ be thelocal 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 theintegers $\lambda_{i,j}(A)$ (defined in \cite{l1}), called the Lyubezniknumbers of $A$, depend only on $X$, but not on the embedding. In this paper, wemake a big step toward a positive answer to this question by proving that inpositive characteristic, for a fixed $X$, the Lyubezink numbers$\lambda_{i,j}(A)$ of the local ring $A$, can only achieve finitely manypossible values under all choices of embeddings.

On Lyubeznik numbers of projective schemes