reesIdeal M
reesIdeal(M,f)
This routine gives the user a choice between two methods for finding the defining ideal of the Rees algebra of an ideal or module $M$ over a ring $R$: The command reesIdeal(M) computes a versal embedding $g: M\to G$ and a surjection $f: F\to M$ and returns the result of symmetricKernel(gf).
When M is an ideal (the usual case) or in characteristic 0, the same ideal can be computed by an alternate method that is often faster. If the user knows a non-zerodivisor $a\in{} R$ such that $M[a^{-1}$ is a free module (for example, when M is an ideal, any non-zerodivisor $a \in{} M$ then it is often much faster to compute reesIdeal(M,a) which computes the saturation of the defining ideal of the symmetric algebra of M with respect to a. This gives the correct answer even under the slightly weaker hypothesis that $M[a^{-1}]$ is of linear type. (See also isLinearType.)
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The following example shows how we handle degrees
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Investigating plane curve singularities:
Proj of the Rees algebra of I \subset{} R is the blowup of I in spec R. Thus the Rees algebra is a basic construction in resolution of singularities. Here we work out a simple case:
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The total transform of the cusp contains the exceptional divisor with multiplicity two. The strict transform of the cusp is a smooth curve but is tangent to the exceptional divisor
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This shows that the strict transform is smooth.
The object reesIdeal is a method function with options.