By Jan Mycielski, Pavel Pudlak, Alan S. Stern

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Q3 Q4 48 4. SINGULARITIES OF MULTIPLICITY EQUAL TO DEGREE DIVIDED BY TWO Subtract β2 Co1 from Co2 , β2 Ro1 from Ro3 , and rename Q3 and Q4 to obtain ⎡ ⎤ Q1 Q3 ⎣Q2 β1 Q1 ⎦ . Q3 Q4 If β1 is zero, then ϕ may be transformed into ϕ(c,μ4 ) . If β1 is not zero, then β1 may be transformed into 1. ) At this point one uses row and column exchanges to transform ϕ into the form of ϕ(∅,μ4 ) . In Case 2C, one starts with ⎤ ⎡ α2 Q2 Q1 ϕ = ⎣Q2 β1 Q1 + β2 Q2 + β3 Q3 ⎦ , Q3 Q4 with β3 = 0. Transform β3 into 1 by multiplying Co2 by β3−1 and renaming Q4 and the constants.

1) The projections T , u ]) BiProj (kk [T PPP ♥♥ PPP ♥ ♥ ♥ PPP ♥ ♥♥ PPP ♥ ♥ v♥♥ ( T ]) u]) k [T Proj (k Proj (kk [u induce isomorphisms uT ) T , u]/I1 (Cu BiProj k [T ❚❚❚❚ ❥ ❥ ❥ ❚❚❚❚π2 π1 ❥❥❥❥ ❚❚❚❚ ❥❥❥❥∼ ∼ ❚❚❚❚ ❥ = = ❥ ❥ t❥❥ * T ]/I2 (C)) u]/I3 (A)) ; k [T Proj (k Proj (kk [u kT ku T] k [u u] in particular, the schemes Proj( Ik2[T (C) ) and Proj( I3 (A) ) are isomorphic. kT T] (2) As a subset of P2 , Proj( Ik2[T (C) ) is equal to {p ∈ C | mp = c}. T , u ] and J = I1 (Cu uT ) Proof. 5 twice. Each time S = k [T T T ).

Notice that gcd(P1 , P2 ) = 1 and gcd(Q3 , Δ) = 1 since I2 (ϕ) has height 2. 9 shows how to modify Q1 and Q2 in order to have gcd(P1 , Q1 ) = 1 and gcd(P2 , Q2 ) = 1. Passing to an aﬃne chart we may assume that p is the origin on the aﬃne curve parametrized by ( gg13 , gg23 ). The blowup of this curve at the origin has two charts, parametrized by ( gg13 , gg21 ) and ( gg23 , gg12 ), respectively. Homogenizing we obtain two curves C and C in P2 parametrized by [g12 : g2 g3 : g1 g3 ] and [g22 : g1 g3 : g2 g3 ], respectively.