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Maximal Degree Subsheaves of Torsion Free Sheaves on Singular Projective Curves

E. Ballico
Transactions of the American Mathematical Society
Vol. 353, No. 9 (Sep., 2001), pp. 3617-3627
Stable URL: http://www.jstor.org/stable/2693805
Page Count: 11
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Maximal Degree Subsheaves of Torsion Free Sheaves on Singular Projective Curves
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Abstract

Fix integers r,k,g with r > k > 0 and g ≥ 2. Let X be an integral projective curve with $g:= p_a(X)$ and E a rank r torsion free sheaf on X which is a flat limit of a family of locally free sheaves on X. Here we prove the existence of a rank k subsheaf A of E such that r(deg(A)) ≥ k(deg(E)) - k(r - k)g. We show that for every g ≥ 9 there is an integral projective curve X, X not Gorenstein, and a rank 2 torsion free sheaf E on X with no rank 1 subsheaf A with 2(deg(A)) ≥ deg(E) - g. We show the existence of torsion free sheaves on non-Gorenstein projective curves with other pathological properties.

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