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We study different notions of slope of a vector bundle over a smoothprojective curve with respect to ampleness and affineness in order to applythis to tight closure problems. This method gives new degree estimates fromabove and from below for the tight closure of a homogeneous $R_+$-primary idealin a two-dimensional normal standard-graded algebra $R$ in terms of the minimaland the maximal slope of the sheaf of relations for some ideal generators. Ifmoreover this sheaf of relations is semistable, then both degree estimatescoincide and we get a vanishing type theorem.

Slopes of vector bundles on projective curves and applications to tight closure problems