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Published online by Cambridge University Press: 06 July 2010
This is a brief exposition of canonical subalgebra bases, their uses and their computation.
From the algebraic point of view, canonical bases are very interesting. Forinstance, if in>R is finitely generated, the study of in>R is simpler than that ofR, and in many cases both algebras share the same properties. As an example,in [Conca et al. 1996] it is shown that if in>R is normal, Cohen–Macaulay, andhas rational singularities, R has the same properties.
From the geometric perspective, SAGBI bases also offer interesting possibili-ties. When in>R is finitely generated, it can be regarded as the associated gradedring of a suitable degree filtration of R. As a consequence in>R can be inter-preted as the special fiber of a one-parameter family with R as a general fiber.In this case the general fiber and the special fiber of the family share geometricproperties. See [Conca et al. 1996; Sturmfels 1996] and also Section 6 below fordiscussion.
This philosophy appears, in the analytic case, in [Teissier 1975] and [Goldinand Teissier 2000], as an approach to resolution of singularities of plane curves:Given a suitable parametrization of a plane curve, construct a flat family ofcurves in such a way that the general fiber is isomorphic to the original curve,and the special fiber is a monomial curve. Then a toric resolution of singularitiesof the special fiber induces a resolution of the generic fiber [Goldin and Teissier2000, § 6].
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