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By John Casey

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Extra info for A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples

Example text

Another important notion is the notion of the HarderNarasimhan polygon which can also be found in Shatz' paper [236]. 1. 4 is due to Flenner [63]. For other results concerning bundles on projective spaces see [211]. - S-equivalence was again first defined for bundles on curves by Seshadri [233]. There, two Sequivalent sheaves are called strongly equivalent. -Langton defined the socle and the extended socle in [135]. For another reference see the paper of Mehta and Ramanathan [176]. 2 can be found in Mumford's book [191].

By definition, a quotient module of Vis an equivalence class of epimorphisms q : V -t F of coherent Os-sheaves, where two epimorphisms qi: V -t Fi, i = 1, 2, are equivalent, ifker(q1 ) = ker(q2), or, equivalently, if there is an isomorphism cp : F 1 -t F2 withq2 = cpoq1 . Here and in the following, quotient modules are used rather than submodules because the tensor product is a right exact functor, so that surjectivity of a homomorphism of coherent sheaves is preserved under base change, whereas injectivity is not.

Moreover, the problem is local in X and S. One may therefore assume that S = Spec(A) for some Noetherian integral domain A with quotient field K, that X = Spec( B) for some finitely generated A-algebra, that A ---t B is injective, and that F = M~ for some finite B-module M. This module M has a finite filtration by B-submodules with factors of the form M; ~ B / p; for prime ideals p; C B. It suffices to consider these factors separately, so that we may further reduce to the case that Jvf = B is integral and A ---t B injective.

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