If $M$ is an ideal or module over a ring $R$, and $F\to M$ is a surjection from a free module, then reesAlgebra(M) returns the ring $Sym(F)/J$, where $J = reesIdeal(M)$.
In the following example, we find the Rees Algebra of a monomial curve singularity. We also demonstrate the use of reesIdeal, symmetricKernel, isLinearType, normalCone, normalCone, specialFiberIdeal.
i1 : S = QQ[x_0..x_3]
o1 = S
o1 : PolynomialRing
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i2 : i = monomialCurveIdeal(S,{3,7,8})
2 2 3 3 3 2 2 5 4 5 3
o2 = ideal (x x - x x , x x - x x , x x - x x , x - x x , x - x x x )
0 2 1 3 1 2 0 3 1 2 0 3 2 1 3 1 0 2 3
o2 : Ideal of S
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i3 : I = reesIdeal i;
o3 : Ideal of S[w ..w ]
0 4
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i4 : reesIdeal(i, Variable=>v)
3 2 2 2 2
o4 = ideal (x x v - x v + x v , x v + x v - x v , x x v - x v + x v ,
1 2 0 0 1 3 2 3 0 2 1 1 3 0 3 0 1 1 2 2
------------------------------------------------------------------------
2 2 3 3 2 2
x x v - x v + x v , x v + x x v - x v , x v - x x v + x v , x x v
0 3 0 1 2 2 4 2 0 1 3 1 0 3 1 0 0 2 2 3 4 2 3 0
------------------------------------------------------------------------
2 2 2 2 2 2 2
- x v + v v , x x x v - x v + v v , (x x x + x x )v - x x v v +
1 1 2 3 0 1 3 0 2 2 1 4 0 2 3 1 3 0 1 2 1 2
------------------------------------------------------------------------
v v )
3 4
o4 : Ideal of S[v ..v ]
0 4
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i5 : I=reesIdeal(i,i_0);
o5 : Ideal of S[w ..w ]
0 4
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i6 : (J=symmetricKernel gens i);
o6 : Ideal of S[w ..w ]
0 4
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i7 : isLinearType(i,i_0)
o7 = false
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i8 : isLinearType i
o8 = false
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i9 : reesAlgebra (i,i_0)
S[w ..w ]
0 4
o9 = ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
3 2 2 2 2 2 2 3 3 2 2 2 2 2 2 2 2 2
(x x w - x w + x w , x w + x w - x w , x x w - x w + x w , x x w - x w + x w , x w + x x w - x w , x w - x x w + x w , x x w - x w + w w , x x x w - x w + w w , (x x x + x x )w - x x w w + w w )
1 2 0 0 1 3 2 3 0 2 1 1 3 0 3 0 1 1 2 2 0 3 0 1 2 2 4 2 0 1 3 1 0 3 1 0 0 2 2 3 4 2 3 0 1 1 2 3 0 1 3 0 2 2 1 4 0 2 3 1 3 0 1 2 1 2 3 4
o9 : QuotientRing
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i10 : trim ideal normalCone (i, i_0)
2 2 3 3 3 2 2 5 4 5 3
o10 = ideal (x x - x x , x x - x x , x x - x x , x - x x , x - x x x )
0 2 1 3 1 2 0 3 1 2 0 3 2 1 3 1 0 2 3
S[w ..w ]
0 4
o10 : Ideal of ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
3 2 2 2 2 2 2 3 3 2 2 2 2 2 2 2 2 2
(x x w - x w + x w , x w + x w - x w , x x w - x w + x w , x x w - x w + x w , x w + x x w - x w , x w - x x w + x w , x x w - x w + w w , x x x w - x w + w w , (x x x + x x )w - x x w w + w w )
1 2 0 0 1 3 2 3 0 2 1 1 3 0 3 0 1 1 2 2 0 3 0 1 2 2 4 2 0 1 3 1 0 3 1 0 0 2 2 3 4 2 3 0 1 1 2 3 0 1 3 0 2 2 1 4 0 2 3 1 3 0 1 2 1 2 3 4
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i11 : trim ideal associatedGradedRing (i,i_0)
2 2 3 3 3 2 2 5 4 5 3
o11 = ideal (x x - x x , x x - x x , x x - x x , x - x x , x - x x x )
0 2 1 3 1 2 0 3 1 2 0 3 2 1 3 1 0 2 3
S[w ..w ]
0 4
o11 : Ideal of ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
3 2 2 2 2 2 2 3 3 2 2 2 2 2 2 2 2 2
(x x w - x w + x w , x w + x w - x w , x x w - x w + x w , x x w - x w + x w , x w + x x w - x w , x w - x x w + x w , x x w - x w + w w , x x x w - x w + w w , (x x x + x x )w - x x w w + w w )
1 2 0 0 1 3 2 3 0 2 1 1 3 0 3 0 1 1 2 2 0 3 0 1 2 2 4 2 0 1 3 1 0 3 1 0 0 2 2 3 4 2 3 0 1 1 2 3 0 1 3 0 2 2 1 4 0 2 3 1 3 0 1 2 1 2 3 4
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i12 : trim specialFiberIdeal (i,i_0)
o12 = ideal (w w , w w , w w )
3 4 1 4 2 3
o12 : Ideal of QQ[w ..w ]
0 4
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