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I.2.6

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This problem has Hartshorne height 2 and Hartshorne volume 3 (I.1.10 > I.1.6).

Results To Utilize

  • Proposition 1.7: \dim A(Y) = \dim Y for a variety Y.
  • Thm 1.8A: \dim B = tr.deg. of Frac(B) over k when B if finitely generated k-algebra and a domain;  ht\ P + \dim B/P = \dim B.
  • Proposition 1.10: Y quasi-affine, then \dim Y = \dim \overline{Y}
  • Proof of Proposition 2.2, basically the part of about the dehomoginization and homogenization maps \alpha and \beta

Outline

  • Use homogenization/dehomoginization to establish that A(Y_i) \cong degree 0 part of S(Y)_{x_i} and ultimately conclude A(Y_i)[x_i,x_i^{-1}] \cong S(Y)_{x_i}.
  • Argue \dim S(Y) = \dim S(Y)_{x_i}.
  • Use 1.7, 1.8A, 1.10 to calculate \dim S(Y)_{x_i} in terms of \dim Y_i.
  • Use that that Y_i form a covering and use I.1.10 to finish the problem.

Conclusions

  • From the last step of the outline, it can be concluded that \dim Y = \dim Y_i whenever Y_i is nonempty.

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