Show that the following two singularities have the same multiplicity, and the same configuration of infinitely near singular points with the same multiplicities, hence the same \(\delta_P\), but are not equivalent.
\(x^4-x y^4=0\).
\(x^4-x^2 y^3-x^2 y^5+y^8=0\).
(Erratum)
As transcribed, the two displayed singularities do not have the same multiplicities at their infinitely near singular points. After blowing up the origin in their common tangent direction, (a) has multiplicity \(2\) and (b) has multiplicity \(3\). In fact their delta invariants are \(7\) and \(9\). Thus the claimed example cannot be proved with the equations printed above. The computation is given below so that the defect is explicit rather than silently repaired to an unverified equation.
Solution
Write \[ C_a:\quad f=x^4-xy^4=0 \] and \[ C_b:\quad g=x^4-x^2y^3-x^2y^5+y^8=0. \]
1. Both singularities have multiplicity \(4\) at the origin and the sameunique tangent line\[ x=0. \]
For both equations the lowest nonzero homogeneous term is \[ x^4. \] Hence both local equations have order \(4\) in the maximal ideal \((x,y)\), and their tangent cone is the quadruple line \(x=0\).
2. Blow up the origin in the chart\[ x=uv, \qquad y=v. \]For (a), the strict transform is\[ \boxed{u(u^3-v)=0}. \]
Substitution gives \[ x^4-xy^4 = u^4v^4-uv^5 = v^4u(u^3-v). \] After removing the exceptional factor \(v^4\), the strict transform is as displayed. Its unique point on the exceptional curve \(E_1:(v=0)\) is \[ Q=(u,v)=(0,0). \]
3. The infinitely near point \(Q\) of (a) has multiplicity\[ \boxed{2}. \]
The strict-transform equation is \[ u^4-uv. \] Its lowest nonzero homogeneous term is \[ -uv, \] of degree \(2\). Thus \[ \mu_Q(C_{a,1})=2. \] The two strict branches are already smooth at \(Q\): they are the line \(u=0\) and the branch \(v=u^3\), with distinct tangent directions \(u=0\) and \(v=0\).
4. After one more blowup at \(Q\), the two strict branches of (a) separate,so there are no further singular points of the strict transform.
The tangent directions at \(Q\) are distinct. Under the blowup of \(Q\), points of the new exceptional curve parametrize these tangent directions, so the two branches meet the new exceptional curve at distinct points. Each branch was already nonsingular before this blowup, hence their strict transforms remain nonsingular and disjoint. Further blowups may be used to make the total transform simple normal crossings with the older exceptional curve, but they occur at smooth points of the curve and contribute no further singular multiplicities.
5. Therefore\[ \boxed{\delta_0(C_a)=7}. \]
By the multiplicity formula D-CRVPLSING, the only singular strict transform multiplicities are \[ 4,2. \] Hence \[ \delta_0(C_a) = \binom42+\binom22 = 6+1 =7. \]
6. Under the same first blowup, the strict transform of (b) is\[ \boxed{ u^4-u^2v-u^2v^3+v^4=0.} \]
Substitution gives \[ \begin{aligned} g(uv,v) &=u^4v^4-u^2v^5-u^2v^7+v^8\\ &=v^4\bigl(u^4-u^2v-u^2v^3+v^4\bigr). \end{aligned} \] Removing the exceptional factor \(v^4\) gives the displayed strict transform. On \(E_1:(v=0)\) its equation is \(u^4=0\), so again its unique point over the origin is \(Q=(0,0)\).
7. The first infinitely near point of (b) has multiplicity\[ \boxed{3}, \]not \(2\).
At \((u,v)=(0,0)\) the lowest-degree nonzero term of \[ u^4-u^2v-u^2v^3+v^4 \] is \[ -u^2v, \] of total degree \(3\). Thus \[ \mu_Q(C_{b,1})=3. \] This already contradicts the assertion that the two singularities have the same infinitely near multiplicities.
8. Blowing up this multiplicity-\(3\) point resolves all singularities ofthe strict transform of (b).
The tangent cone at \(Q\) is \[ u^2v=0, \] with tangent directions \(v=0\) and \(u=0\).
For the direction \(v=0\), use \(v=uw\). Substitution into the strict-transform equation gives, after removing the exceptional factor \(u^3\), \[ u-w-u^2w^3+uw^4=0. \] At \((u,w)=(0,0)\) this is nonsingular because its linear part is \(u-w\).
For the direction \(u=0\), use \(u=vz\). After removing \(v^3\) one obtains \[ vz^4-z^2-v^2z^2+v=0. \] At \((v,z)=(0,0)\) this is nonsingular because its linear part is \(v\).
These are the two points of the new exceptional divisor determined by the two tangent directions. Thus the strict transform has no singular point after this second blowup.
9. Therefore\[ \boxed{\delta_0(C_b)=9}. \]
The singular multiplicities encountered are \[ 4,3. \] Hence D-CRVPLSING gives \[ \delta_0(C_b) = \binom42+\binom32 = 6+3 =9. \]
10. The transcribed claim of V.3.8 is therefore false.
Although the two original singularities both have multiplicity \(4\), their first infinitely near singular points have different multiplicities: \[ 2\ne3. \] Their delta invariants also differ: \[ 7\ne9. \] Consequently they cannot have the same configuration with the same multiplicities, and in particular they are not equivalent in the sense of (3.9.4). The final inequivalence assertion is true, but the stronger claimed matching of resolution data is not true for the equations transcribed on this card.
11. Q.E.D. for the transcribed statement: the displayed equations give acountercalculation to its claimed matching data.
Steps <1>1–<1>5 compute the infinitely near multiplicities and delta invariant of (a); steps <1>6–<1>9 do the same for (b); step 10 exhibits the contradiction to the source claim.