Let \(\varphi\) be the quadratic transformation of \((4.2.3)\), centered at \(P_1, P_2, P_3\). If \(C\) is an irreducible curve of degree \(d\) in \(\PP^2\), distinct from the three lines joining pairs of \(P_1,P_2,P_3\), and with points of multiplicity \(r_1, r_2, r_3\) at \(P_1, P_2, P_3\), then the strict transform \(C^{\prime}\) of \(C\) by \(\varphi\) has degree \[ d^{\prime}=2 d-r_1-r_2-r_3 \] and has points of multiplicity
\(d-r_2-r_3\) at \(Q_1\),
\(d-r_1-r_3\) at \(Q_2\) and
\(d-r_1-r_2\) at \(Q_3\).
The curve \(C\) may have arbitrary singularities.
Hint: Use (Ex. 3.2).
(Erratum)
The exclusion of the three lines joining pairs of base points is necessary. Those are precisely the three curves contracted by the quadratic transformation. For example, the line through \(P_2\) and \(P_3\) has \[ d=1, \qquad r_1=0, \qquad r_2=r_3=1, \] so the displayed degree formula gives \(d'=0\): its image is the point \(Q_1\), not a plane curve.
Solution
Let \[ p:S=\operatorname{Bl}_{P_1,P_2,P_3}\PP^2\longrightarrow\PP^2 \] be the blowup of the three base points. Write \(H\) for the pullback of a line and \(E_i\) for the exceptional curve over \(P_i\).
1. The quadratic transformation lifts to a morphism\[ q:S\longrightarrow\PP^{2\prime} \]defined by the complete linear system\[ |H'|, \qquad H'=2H-E_1-E_2-E_3. \]
After choosing coordinates so that the \(P_i\) are the three coordinate vertices, the quadratic transformation is \[ [x_0:x_1:x_2] \dashrightarrow [x_1x_2:x_0x_2:x_0x_1] \] P-AGH46CREMONA. Its defining quadrics are exactly the conics through all three base points, so on the blowup their strict transforms have divisor class \[ 2H-E_1-E_2-E_3. \] The three base points are removed by the blowup, hence this system defines the morphism \(q\) resolving the rational map.
2. Let\[ L_{23}=H-E_2-E_3, \qquad L_{13}=H-E_1-E_3, \qquad L_{12}=H-E_1-E_2 \]be the strict transforms of the three lines joining pairs of base points.Then \(q\) contracts them respectively to three points\[ Q_1, \qquad Q_2, \qquad Q_3, \]and identifies \(S\) with\[ \operatorname{Bl}_{Q_1,Q_2,Q_3}\PP^{2\prime}. \]
For example, \[ H'\cdot L_{23} = (2H-E_1-E_2-E_3)\cdot(H-E_2-E_3) = 2-1-1 =0. \] The same calculation holds for the other two lines. Each \(L_{jk}\) is a nonsingular rational curve with \[ L_{jk}^2=1-1-1=-1, \] so the resolved quadratic transformation contracts precisely these three \((-1)\)-curves.
The quadratic transformation is an involution P-AGH46CREMONA. Applying the same construction on the target therefore blows up the three target base points \(Q_1,Q_2,Q_3\) and recovers \(S\). Thus \(L_{23},L_{13},L_{12}\) are the exceptional curves over \(Q_1,Q_2,Q_3\), respectively.
3. The strict transform \(\overline C\subseteq S\) of \(C\) has class\[ \boxed{ \overline C \sim dH-r_1E_1-r_2E_2-r_3E_3.} \]
The total transform of a plane curve of degree \(d\) has class \(dH\). At a blown-up point \(P_i\) of multiplicity \(r_i\), the exceptional curve occurs in the total transform with coefficient \(r_i\). Therefore the standard strict-transform formula gives the displayed class. This calculation uses only the multiplicities at the three chosen points; \(C\) may have arbitrary other singularities.
4. Because \(C\) is not one of the three contracted lines, the morphism \(q\) maps \(\overline C\) birationally onto the target strict transform \(C'\subseteq\PP^{2\prime}\).
The morphism \(q\) is an isomorphism away from the three curves \(L_{23},L_{13},L_{12}\). Since \(C\) is irreducible and is not one of the corresponding three lines in the source plane, its strict transform \(\overline C\) is not one of these exceptional curves. Hence its generic point lies in the locus on which \(q\) is an isomorphism. Its image is therefore an irreducible curve \(C'\), and \[ q|_{\overline C}:\overline C\dashrightarrow C' \] is birational.
5. The degree of \(C'\) is\[ \boxed{ d'=2d-r_1-r_2-r_3.} \]
The pullback by \(q\) of a line in the target plane is \(H'\). Therefore \[ \deg C' = H'\cdot\overline C. \] Using steps 1 and 3 together with \[ H^2=1, \qquad E_i^2=-1, \qquad H\cdot E_i=E_i\cdot E_j=0\quad(i\ne j), \] gives \[ \begin{aligned} d' &=(2H-E_1-E_2-E_3) \cdot(dH-r_1E_1-r_2E_2-r_3E_3)\\ &=2d-r_1-r_2-r_3. \end{aligned} \]
6. The multiplicity of \(C'\) at \(Q_1\) is\[ \boxed{ \mu_{Q_1}(C')=d-r_2-r_3.} \]
Under \[ q:S=\operatorname{Bl}_{Q_1,Q_2,Q_3}\PP^{2\prime} \longrightarrow\PP^{2\prime}, \] the exceptional curve over \(Q_1\) is \(L_{23}\). For the blowup of a point, the multiplicity of a curve at the centre equals the intersection of its strict transform with the exceptional curve. Hence \[ \mu_{Q_1}(C') = \overline C\cdot L_{23}. \] By steps <1>2–<1>3, \[ \begin{aligned} \overline C\cdot L_{23} &=(dH-r_1E_1-r_2E_2-r_3E_3) \cdot(H-E_2-E_3)\\ &=d-r_2-r_3. \end{aligned} \]
7. Similarly,\[ \boxed{ \mu_{Q_2}(C')=d-r_1-r_3, \qquad \mu_{Q_3}(C')=d-r_1-r_2.} \]
The exceptional curves over \(Q_2\) and \(Q_3\) are respectively \[ L_{13}=H-E_1-E_3 \] and \[ L_{12}=H-E_1-E_2. \] Intersecting each with the class of \(\overline C\) from step 3 gives \[ \overline C\cdot L_{13}=d-r_1-r_3 \] and \[ \overline C\cdot L_{12}=d-r_1-r_2. \] By the same blowup multiplicity formula as in step 6, these are exactly the target multiplicities at \(Q_2,Q_3\).
8. The formulas remain valid when \(C\) has arbitrary singularities away from, or at, the three base points.
Every calculation above takes place in the divisor class group of the smooth common resolution \(S\) and uses only the multiplicities \(r_i\) entering the strict-transform class. No step assumes that the germ of \(C\) at any \(P_i\) is ordinary or nonsingular, nor that \(C\) is nonsingular elsewhere. Thus the degree and target-multiplicity formulas hold for arbitrary singularities.
9. Q.E.D.
Step 5 proves the degree formula, steps <1>6–<1>7 prove the three multiplicity formulas, and step 8 establishes the stated generality.