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Slopes of vector bundles on projective curves and applications to tight closure problems. (English) Zbl 1041.13002
In J. Algebra 265, 45–78 (2003; Zbl 1099.13010), H. Brenner has showed how to connect the tight closure theory of standard graded rings to the theory of projective bundles. In this paper, the author applies his method to two dimensional standard graded rings over a field. The author investigates the minimal and maximal slope of locally free sheaves on smooth projective curves and shows how to apply these invariants to tight closure problems. He describes degree bounds that force elements to belong to the tight closure of a given ideal; similarly, he gives degree bounds that guarantee that an element is not in the tight closure of an ideal unless it is in the ideal itself. While it is known that examples in tight closure theory are generally hard to compute, these results provide a new and large class of examples in tight closure theory obtained from a coherent point view.

MSC:
 13A35 Characteristic $$p$$ methods (Frobenius endomorphism) and reduction to characteristic $$p$$; tight closure 14H60 Vector bundles on curves and their moduli 13A02 Graded rings 14F17 Vanishing theorems in algebraic geometry
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References:
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