On Lyubeznik Numbers of Projective Schemes

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 ι : X →֒ P n k . G. Lyubeznik asked (in [15]) whether the integers λi,j(A) (defined in [14]), 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 λi,j(A) of the local ring A, can only achieve finitely many possible values under all choices of embeddings.