Abstract
In this chapter we assume, basically for simplicity only, that all schemes be projective varieties over a field (of any characteristic unless otherwise stated), which without significant loss of generality may be assumed algebraically closed.
The chapter gives some basic constructions in the category of projective k-varieties: the blowing-up of a closed subscheme and of subbundles. We introduce the Grassmann bundles, and the related construction of a parameter variety for the joining lines for a projective, embedded scheme. The secant variety and the join are given as applications of these constructions.
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References
Eisenbud, D.: Introduction to Commutative Algebra with a View Towards Algebraic Geometry. Graduate Texts in Mathematics. Springer, Berlin (1997)
Fulton, W.: Intersection Theory, 1st edn. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 2. Springer, Berlin (1984), 2nd edn. (1998)
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© 2012 Springer-Verlag Berlin Heidelberg
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Holme, A. (2012). Some Basic Constructions in the Category of Projective k-Varieties. In: A Royal Road to Algebraic Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19225-8_21
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DOI: https://doi.org/10.1007/978-3-642-19225-8_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-19224-1
Online ISBN: 978-3-642-19225-8
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