The Rees Algebra of an ideal I appeared classically as the bihomogeneous coordinate ring of the blow up of the ideal I, used in resolution of singularities. Though the general case is still out of reach, we illustrate with some simple examples of plane curve singularities.
First the cusp in the affine plane
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The cusp is singular at the maximal ideal (x,y), so we blow that up, and examine the ``total transform'', that is, the ideal generated by the x^2-y^3 in the Rees algebra.
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Application of first flattenRing serves to make B a quotient of the polynomial ring T in 4 variables; otherwise it would be a quotient of R[w_0,w_1], which Macaulay2 treats as a polynomial ring in 2 variables, and the calculation of the singular locus later on would be wrong.
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We see that the reduced preimage consists of two codimension 1 components, the `exceptional divisor', which is the pullback of the point we blew up, (x,y), and the `strict transform'. The two components meet in a double point in the 2 dimensional variety B \subset{} A^2\times P^1. We have to saturate with respect to the irrelevant ideal to understand what's going on.
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We can see the multiplicities of these components by comparing their degrees to the degrees of the reduced components
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That is, the exceptional component occurs with multiplicity 2 (in general we'd get the exceptional component with multiplicity equal to the multiplicity of the singular point we blew up.)
We next investigate the singularity of the strict transform. We want to see it as a curve in P^1 x A^2, that is, as an ideal of T = kk[w_0,w_1,x,y]
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We see that the singular locus of the strict transform is empty; that is, the curve is smooth.
We could have made the computation in B as well:
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Next we look at the desingularization of a tacnode; it will take two blowups.
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Here proj1 mm is the ideal of the exceptional divisor. The strict transform is, by definition, obtained by saturating it away, The strict transform of the tacnode is not yet smooth: it consists of two smooth branches, meeting transversely at a point:
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We compute the singular point of the strict transform:
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...and blow up B1, getting a variety in P^2 x P^1 x A^2
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We compute the singular locus once again:
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The answer, ideal 1 shows that the second blowup desingularizes the tacnode.
It is not necessary to repeatedly blow up closed points: there is always a single ideal that can be blown up to desingularize (Hartshorne, Algebraic Geometry,Thm II.7.17). In this case, blowing-up (x,y^2) desingularizes the tacnode x^2-y^4 in a single step.
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So this single blowup is already nonsingular.
The object PlaneCurveSingularities is a symbol.