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I have met the notion of Grassmannian bundle of a vector bundle over a variety in intersection theory, but anywhere I look I just find a brief recall of how the stalks look like (my references so far are "3264 and all that" by Eisenbud and Harris, "Intersection Theory" by Fulton). I was wondering whether there is any good reference to see the actual definition and construction and the universal properties of this object...

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    $\begingroup$ It depends on exactly what you want, but this can often be found in books on algebraic topology (e.g. Hatcher's book on K-theory). $\endgroup$ – Paul Siegel Jan 27 '14 at 21:44
  • $\begingroup$ Something analogous to the standar introduction to grassmannians, such as scheme structure, universal subbundle... $\endgroup$ – Stefano Jan 27 '14 at 21:54
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    $\begingroup$ math.utah.edu/~bertram/courses/hilbert are the notes a professor of mine referred to when we went over Hilbert schemes and the realization of certain moduli spaces(including Grassmannians) as schemes. To me, the notes seem rather detailed though they are completely void of exercises. Of course, it's probably really useful to see Grassmannians outside of scheme theory (e.g. Paul Siegel's recommendation). $\endgroup$ – PVAL-inactive Jan 27 '14 at 22:13
  • $\begingroup$ Thanks, the references have been very useful! Especially Bertram's notes: I read on "Geometry of Schemes" the construction over an affine scheme and then the idea of "twisted" patching was clear. $\endgroup$ – Stefano Jan 29 '14 at 18:57

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