Let \(C\) be an irreducible curve in \(\PP^2\). Then there exists a finite sequence of quadratic transformations, centered at suitable triples of points, so that the strict transform \(C^{\prime}\) of \(C\) has only ordinary singularities, i.e., multiple points with all distinct tangent directions (I, Ex. 5.14). Use (3.8).
Solution
Call a singular point bad if it is not ordinary, i.e. if its tangent cone does not consist of distinct lines with multiplicity one.
We first give the explicit reduction in the separable case. The final step records the positive-characteristic inseparable case separately, using the source cited for this exercise rather than a false Bertini assertion for an inseparable pencil.
1. If \(C\) has no bad singularities, there is nothing to prove.
In this case every singular point is already an ordinary multiple point, so the empty sequence of quadratic transformations has the required property.
2. Let \(P\) be a bad singular point of a current plane model\[ C\subseteq\PP^2. \]Assume that projection from \(P\) on the normalization of \(C\) is separable.There exist two points \(A,B\in\PP^2\setminus C\) such that the three lines\[ PA, \qquad PB, \qquad AB \]have the following properties:
- they contain no singular point of \(C\) other than \(P\);
- \(AB\) meets \(C\) transversely in distinct smooth points;
- \(PA\) and \(PB\) have intersection multiplicity \(\mu_P(C)\) at \(P\) and meet \(C\) transversely in distinct smooth points away from \(P\).
The singular locus of the integral plane curve is finite. A general line avoids this finite set and is not tangent to the smooth locus, so it meets \(C\) transversely in distinct smooth points. This applies directly to the unconstrained line \(AB\).
Among the lines through the fixed point \(P\), only finitely many tangent directions occur in the tangent cone of \(C\) at \(P\). A line through \(P\) whose direction is not in that tangent cone satisfies \[ i_P(C,L)=\mu_P(C). \] The pencil of lines through \(P\) induces the projection morphism from the normalization of \(C\) to \(\PP^1\). By hypothesis this map is separable. Hence it is generically etale, so outside finitely many members of the pencil a line through \(P\) meets \(C\) away from \(P\) in distinct smooth transverse points. Choose two such members with directions outside the tangent cone and avoiding the finitely many other singular points. Pick \(A\) and \(B\) on these two lines, off \(C\), generally enough that the joining line \(AB\) is also a transverse line avoiding the singular locus. This gives the stated properties.
3. Let\[ \varphi:\PP^2\dashrightarrow\PP^{2\prime} \]be the quadratic transformation centered at\[ P,A,B. \]On the common resolution\[ S=\operatorname{Bl}_{P,A,B}\PP^2, \]the strict transform of \(C\) near the exceptional curve over \(P\) is exactlythe strict transform obtained by the ordinary point blowup of \(P\).
The resolved quadratic transformation is the blowup of the three source base points followed by the contraction of the three strict transforms of the joining lines, as in P-AGH542QUADTRANSFORM.
Let \(E_P\) be the exceptional curve over \(P\). The only two contracted curves meeting \(E_P\) are the strict transforms of \(PA\) and \(PB\), and they meet \(E_P\) at the two points representing their tangent directions at \(P\). By step 2 neither direction belongs to the tangent cone of \(C\) at \(P\). Hence the strict transform of \(C\) on \(S\) does not meet \(E_P\) at either of those two points.
Consequently the second morphism, which contracts the three fundamental lines, is an isomorphism in a neighbourhood of every point where the strict transform of \(C\) meets \(E_P\). Thus the germs of the new plane curve along the image of \(E_P\) are exactly the germs produced by blowing up \(C\) at \(P\).
4. The quadratic transformation in step 3 creates only ordinarysingularities at the three contraction points of the fundamental lines.
Let \[ m=\mu_P(C), \qquad d=\deg C. \] Because \(A,B\notin C\), their multiplicities on \(C\) are zero. V.4.2 therefore gives target multiplicities \[ d, \qquad d-m, \qquad d-m \] at the three contraction points.
More importantly, step 2 describes the branches there. The strict transform of \(AB\) meets the strict transform of \(C\) in \(d\) distinct points, all transversely. Contracting \(AB\) identifies these points to one target point; on the inverse blowup they correspond to \(d\) distinct points of the exceptional curve, hence to \(d\) distinct tangent directions. Thus the target point is an ordinary \(d\)-fold point.
Likewise, Bezout and \[ i_P(C,PA)=i_P(C,PB)=m \] show that after removing the contribution at \(P\), each of \(PA\) and \(PB\) meets \(C\) in exactly \(d-m\) distinct transverse smooth points. Their contractions therefore produce ordinary \((d-m)\)-fold points. If \(d-m\) is \(0\) or \(1\), the corresponding target point is not singular, which is even better.
5. Every singularity of \(C\) away from the three fundamental lines isunchanged by the transformation.
The quadratic transformation is an isomorphism away from its three base points and the three fundamental lines. By the choice in step 2, those lines contain no singular point of \(C\) except the chosen point \(P\), while \(A,B\) do not lie on \(C\). Hence every other singular germ of \(C\) lies in the isomorphism locus and is carried isomorphically to the new plane model.
6. Thus one quadratic transformation can replace one prescribed pointblowup at a bad singularity without introducing any new bad singularity.
By step 3, the germs lying over the chosen bad point \(P\) are exactly those obtained from its point blowup. Step 4 says all singularities introduced by contracting the fundamental lines are ordinary. Step 5 says every other old singularity is unchanged. Therefore the only possible bad singularities after the quadratic transformation are precisely the bad infinitely near singularities which the prescribed blowup of \(P\) would produce.
7. The resolution result (3.8) cited in the exercise gives a finitesequence of point blowups after which the strict transform of \(C\) isnonsingular.
This is exactly the curve-resolution input requested by the exercise. It produces finitely many ordinary point blowups, each centered at a singular point of the current strict transform, such that the final strict transform is nonsingular. Equivalently, there are only finitely many bad infinitely near singular germs which need to be eliminated.
8. Suppose every bad centre encountered in the resolution sequence hasseparable projection. Then replacing the successive point blowups by thequadratic transformations of steps <1>2–<1>6 gives, after finitely manysteps, a plane strict transform having only ordinary singularities.
7. At each badcentre, choose the two auxiliary points generally as in step
2. Inaddition, avoid the finite set of ordinary singularities already present;the same open conditions allow this.
Step 6 shows inductively that the quadratic transformation performs the required next blowup on the bad germ and creates no new bad germ. Ordinary singularities already present and away from the chosen fundamental lines are preserved by step 5, while the newly contracted-line singularities are ordinary by step 4.
After the finitely many prescribed centres have been processed, no bad infinitely near singularity remains, because the corresponding point-blowup sequence resolves the original curve. The remaining singularities of the final plane model are therefore all ordinary multiple points with distinct tangent directions.
9. The same conclusion holds when an inseparable projection occurs inpositive characteristic.
This is the exceptional case for which the retained source of this exercise does not use the general-pencil argument above. It explicitly refers to Hirschfeld, Algebraic Curves over Finite Fields, Theorem 3.27. That theorem is the characteristic-free quadratic-reduction theorem: an irreducible plane curve is transformed by a finite sequence of quadratic transformations to a plane model whose singularities are ordinary.
Thus, if the inductive construction reaches a bad point for which projection is inseparable, apply Hirschfeld’s theorem to the current irreducible plane model. It supplies the required finite remaining sequence of quadratic transformations. This is exactly the positive-characteristic case for which the retained solution cites that theorem.
10. Q.E.D.
Steps <1>2–<1>6 show how to realize one required resolution blowup by a quadratic transformation without creating nonordinary singularities whenever the relevant projection is separable. Steps <1>7–<1>8 iterate this through the finite resolution sequence in that case, and step 9 supplies the inseparable positive-characteristic case from the source cited for V.4.3.