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    Geometry of Elliptic Curves - Joe Silverman

    Then there exists a unique filtration by subbundles. Two vector bundles with isomorphic associated gradeds are called S-equivalent. On higher-dimensional varieties the filtration also always exist and is unique, but the associated graded components may no longer be bundles. For Gieseker stability the inequalities between slopes should be replaced with inequalities between Hilbert polynomials. Narasimhan—Seshadri theorem says that stable bundles on a projective nonsingular curve are the same as those that have projectively flat unitary irreducible connections. For bundles of degree 0 projectively flat connections are flat and thus stable bundles of degree 0 correspond to irreducible unitary representations of the fundamental group.

    Kobayashi and Hitchin conjectured an analogue of this in higher dimensions. It was proved for projective nonsingular surfaces by Donaldson , who showed that in this case a vector bundle is stable if and only if it has an irreducible Hermitian—Einstein connection. A coherent sheaf E is semistable if the following two conditions hold: [1].

    Note that Coh d is a Serre subcategory for any d , so the quotient category exists. There are also other directions for generalizations, for example Bridgeland 's stability conditions. From Wikipedia, the free encyclopedia. Main article: Kobayashi—Hitchin correspondence.

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    Topics in algebraic curves. Five points determine a conic Projective line Rational normal curve Riemann sphere Twisted cubic. Elliptic function Elliptic integral Fundamental pair of periods Modular form. Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell—Weil theorem Nagell—Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof—Elkies—Atkin algorithm. Elliptic curve cryptography Elliptic curve primality.

    De Franchis theorem Faltings's theorem Hurwitz's automorphisms theorem Hurwitz surface Hyperelliptic curve. Dual curve Polar curve Smooth completion.