Singularities of Varieties
Singularities of Varieties
Singularities of Varieties
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Rees Algebras<br />
Let R be the function ring <strong>of</strong> an algebraic or analytic variety (for instance,<br />
R = C[x 1 , ...,x n ]). A Rees algebra over R is a graded ring A = ⊕ ∞ i=0 A i,<br />
such that R = A 0 ⊇ A 1 ⊇ A 2 ⊇ . ...<br />
The Rees algebra is finitely generated iff there exists b such that A is<br />
generated by all elements <strong>of</strong> degree at most b.<br />
The singular set <strong>of</strong> a Rees algebra is the set <strong>of</strong> all points where A i has order<br />
at least i.<br />
Consider a blowup at a center in the singular locus. Then the proper<br />
transform <strong>of</strong> the Rees algebra is defined as the Rees algebra generated by<br />
the controlled transforms <strong>of</strong> (A i ,i) in degree i.<br />
VO AAG 45
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