Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 42 (2001), No. 2, 509-516.

Inflection Points on Real Plane Curves Having Many Pseudo-Lines

Johannes Huisman

Institut Mathématique de Rennes, Université de Rennes 1 Campus de Beaulieu, 35042 Rennes Cedex, France e-mail: huisman@univ-rennes1.fr

Abstract. A pseudo-line of a real plane curve C is a global real branch of C(ℝ) that is not homologically trivial in ℙ^{2}(ℝ). A geometrically integral real plane curve C of degree d has at most d − 2 pseudo-lines, provided that C is not a real projective line. Let C be a real plane curve of degree d having exactly d − 2 pseudo-lines. Suppose that the genus of the normalization of C is equal to d − 2. We show that each pseudo-line of C contains exactly 3 inflection points. This generalizes the fact that a nonsingular real cubic has exactly 3 real inflection points. MSC 2000: 14H45, 14P99 Keywords: real plane curve, pseudo-line, inflection point

1. Introduction

Let C ⊆ ℙ^{2} be a real algebraic plane curve. The set C(ℝ) of real points is a real analytic subset of ℙ^{2}(ℝ) and has a finite number of global branches [1]. Let B be such a branch. Since B is a real analytic subset of ℙ^{2}(ℝ), it has a fundamental class [B] in the homology group H_{1}(ℙ^{2}(ℝ), ℤ/2ℤ) of ℙ^{2}(ℝ) [2]. We say that B is a pseudo-line of the curve C if [B] ≠ 0 (see Figure 1). A convenient and equivalent way to express that B is a pseudo-line is that there is a projective real line L ⊆ ℙ^{2}(ℝ) intersecting B in an odd number of points (counted with multiplicity). In fact, if B is a pseudo-line then any projective real line L ⊆ ℙ^{2}(ℝ) intersects B in an odd number of points (provided, of course, that L ≠ B). These statements all follow from the fact that H_{1}(ℙ^{2}(ℝ), ℤ/2ℤ) is isomorphic to ℤ/2ℤ, that the intersection product on H_{1}(ℙ^{2}(ℝ), ℤ/2ℤ) is nondegenerate, and that the fundamental class [L] of a projective real line L ⊆ ℙ^{2}(ℝ) is nonzero in H_{1}(ℙ^{2}(ℝ), ℤ/2ℤ) [8].

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag

Figure 1. A real plane curve having 4 real branches, exactly 3 of them are pseudo-lines. The marked points are the real inflection points of the curve.

Real plane curves with many pseudo-lines have rarely been studied. The reason may be that such curves are necessarily singular. Indeed, if C ⊆ ℙ^{2} is a real algebraic curve having many pseudo-lines then two distinct pseudo-lines B and B′ of C intersect each other in a singular point of C (since [B] · [B′] ≠ 0). Curves with many pseudo-lines seem interesting to study because they turn out to have a more uniform behavior than, for example, nonsingular plane curves. In fact, one may think of the class of curves having many pseudo-lines and the class of nonsingular plane curves as lying at opposite ends on the spectrum of all plane curves of given degree. A study of the passage from curves having many pseudo-lines to curves having less pseudo-lines might shed a different light on the geometry of nonsingular real plane curves. In this paper we study the geometry of curves with many pseudo-lines.

Let us make precise what we mean by real plane curves having many pseudo-lines. Let C be a geometrically integral real plane curve, i.e., its complexification C ×_{ℝ} ℂ is reduced and irreducible [5]. Let d be the degree of C. We say that C has many pseudo-lines if C has exactly d − 2 pseudo-lines and if the genus of the normalization of C is equal to d − 2 (Figure 1 is a picture of a curve of degree 5 having many pseudo-lines). In Section 2 we give a motivation for the present definition of a real plane curve having many pseudo-lines.

There are many examples of real plane curves having many pseudo-lines: all nonsingular real conics and all nonsingular real cubics have many pseudo-lines. In fact, there are curves having many pseudo-lines of arbitrary degree d ≥ 2. Indeed, choose a nonsingular real conic X in ℙ^{2} and choose d − 2 real projective lines L_{1}, …, L_{d−2} in ℙ^{2} in general position such that L_{i}(ℝ) does not intersect X(ℝ) for i = 1, …, d − 2. Let C′ be the union of X and L_{1}, …, L_{d−2}. Then, by a well known result of Brusotti one can “deform away” all the nonreal singularities of the real plane curve C′. What one gets is a real plane curve C of degree d having exactly d − 2 pseudo-lines such that the genus of the normalization of C is equal to d − 2, i.e., C is a real plane curve of degree d having many pseudo-lines (see [6] for another proof of the existence of such curves).

The paper is devoted to the study of real inflection points on real plane curves having many pseudo-lines. The main result is the following:

Theorem 1. Let C be a real plane curve having many pseudo-lines. Then, each pseudo-line of C contains exactly 3 inflection points.

Theorem 1 generalizes that what is known for nonsingular real cubics [9]: A nonsingular real cubic has a unique pseudo-line which admits the structure of a real Lie group isomorphic to the circle group. The set of inflection points on the pseudo-line coincide with the 3-torsion subgroup. The latter subgroup is isomorphic to ℤ/3ℤ. Therefore, the pseudo-line contains exactly 3 inflection points.

Using Bezout’s Theorem and the fact that a real projective line necessarily intersects a pseudo-line, it can easily be seen that a real branch of C that is not a pseudo-line cannot contain inflection points. Hence, Theorem 1 implies that C has exactly 3(d−2) real inflection points. This total of 3(d − 2) real inflection points can also be obtained from the generalized Klein Equation [10, 11]. However, that equation does not imply anything concerning the distribution of the real inflection points over the different real branches.

In order to prove Theorem 1, one may show that a smooth pseudo-line of any real plane curve contains at least 3 inflection points. Then, applying the generalized Klein Equation, one deduces that each pseudo-line of C contains exactly 3 inflection points. However, we present here a proof which does not use the generalized Klein Equation. The proof is inspired on the case of a nonsingular real cubic, in spite of the absence of a natural structure of a Lie group on a pseudo-line of C.

Theorem 1 has already been proved in [6] in the case where the curve C has, besides the d − 2 pseudo-lines, yet another real branch. The proof in [6] does not seem to generalize to the case of curves having many pseudo-lines.

Acknowledgement. I thank J.-J. Risler and the referee for their remarks on an earlier version of the paper.

2. The number of pseudo-lines of a real plane curve

In this section we briefly justify the present definition of a real curve having many pseudolines.

Proposition 2. Let C ⊆ ℙ^{2} be a geometrically integral real plane curve of degree d. If C has at least d − 1 pseudo-lines then C is a real projective line in ℙ^{2}.

Proof. Let C̃ be the normalization of C and let g̃ be its genus. By the genus formula,

g̃ = \frac{1}{2}(d − 1)(d − 2) − μ,

where μ is the multiplicity of the singular locus of C [3]. By hypothesis, C has at least d − 1 pseudo-lines. Since any two distinct pseudo-lines of C intersect each other, μ is greater than or equal to \frac{1}{2}(d − 1)(d − 2). Hence, g̃ = 0 and C is a rational curve. Then, C can have at most 1 pseudo-line. It follows that d−1 ≤ 1, i.e., d = 1 or d = 2. Then, C is a real projective line or a real conic. But a geometrical integral real conic has no pseudo-lines. Since C is supposed to have at least d−1 pseudo-lines, C is not a conic. Therefore, C is a real projective line in ℙ^{2}.

By the above proposition, real plane curves of degree d having at least d − 1 pseudo-lines do not constitute an interesting class of real curves as far as geometry is concerned. Therefore, we concentrate on real plane curves having exactly d − 2 pseudo-lines:

Proposition 3. Let C ⊆ ℙ^{2} be a geometrically integral real plane curve of degree d having exactly d − 2 pseudo-lines. Then, the genus g̃ of the normalization of C is equal to d − 3 or d − 2. Moreover, one has g̃ = d − 2 if and only if any two distinct pseudo-lines of C intersect in one point only—the intersection being transverse—and C has no other singularities.

Proof. Let C̃ be the normalization of C. By the genus formula [3], the genus g̃ of C̃ satisfies

g̃ ≤ \frac{1}{2}(d − 1)(d − 2) − \frac{1}{2}(d − 2)(d − 3) = d − 2.

Equality holds if and only if any two distinct pseudo-lines of C intersect in one point only—the intersection being transverse—and C has no other singularities. Note that, by definition, the intersection of two distinct real branches of C is transverse in a point P if and only if both branches are smooth at P and both tangent lines are distinct.

Harnack’s Inequality [4] states that the number s of real branches of C satisfies the inequality

s ≤ g̃ + 1.

Since C has at least d − 2 real branches, one has d − 2 ≤ s ≤ g̃ + 1, i.e., d − 3 ≤ g̃. Therefore, g̃ is equal to d − 3 or d − 2.

In view of the two preceding propositions, the definition of a real plane curve having many pseudo-lines now seems reasonable.

3. Inflection points on pseudo-lines

We fix, throughout this section, an integer d ≥ 2 and a geometrically integral curve C ⊆ ℙ^{2} of degree d having many pseudo-lines. In particular, C has exactly d − 2 pseudo-lines. Let ν: C̃ → C be the normalization of C. By hypothesis, the genus g̃ of C̃ is equal to d − 2.

Let us collect in the following statement some immediate properties satisfied by C, some of which have already been mentioned above:

Proposition 4.

1. The curve C has either d − 2 or d − 1 real branches.

2. Two distinct pseudo-lines of C intersect in one point only. These singularities are the only singularities of C. They are all real ordinary multiple points. In particular, each real branch of C is a smooth real analytic curve in ℙ^{2}(ℝ).

3. The curve C has only ordinary real inflection points and no real multitangent lines.

Proof. 1. By Harnack’s Inequality [4] the number s of real branches of C satisfies s ≤ g̃+1 = d − 1. Since C has many pseudo-lines, s ≥ d − 2. Therefore, s = d − 2 or d − 1.

2. These properties have already been shown above (cf. Proposition 3).

3. Suppose that there is a real branch B of C containing a nonordinary real inflection point P. Let L ⊆ ℙ^{2} be a real projective line such that L(ℝ) is tangent to B at P. By hypothesis, the order of contact of L at P is at least 4. Since L(ℝ) intersects at least d − 3 pseudo-lines of C different from B, the degree of the intersection product L · C is at least 4 + (d − 3) = d + 1. Contradiction by Bezout’s Theorem since C is of degree d. Therefore, C does not have nonordinary real inflection points.

Let us make precise what we mean by real multitangent lines. Let L ⊆ ℙ^{2} be a real line. We say that L is a multitangent line of C if either L is tangent to C at a nonreal closed point, or L is tangent to C at two distinct real points. This is in accordance with the fact that a nonreal closed point is a point of degree 2.

Suppose that L is a multitangent line of C that is tangent to C at a nonreal point P. Since L(ℝ) intersects each of the d − 2 pseudo-lines of C, the degree of L · C is at least (d − 2) + 2 deg(P) = d + 2. This contradicts the fact that L · C is of degree d. Hence, there are no multitangent lines of C that are tangent at a nonreal point.

Suppose that L is a multitangent line of C that is tangent to C at two distinct real points P and P′. Let B (resp. B′) be the real branch of C to which L(ℝ) is tangent at P (resp. P′). There are 3 cases to consider:

1. B and B′ are distinct pseudo-lines of C,

2. B and B′ are one and the same pseudo-line of C, and

3. B or B′ is not a pseudo-line of C. We show that each of these cases leads to a contradiction. In each of the first two cases, the main observation is that L(ℝ) intersects each pseudo-line of C in an odd number of points.

In case 1, the degrees of L(ℝ) · B and L(ℝ) · B′ are at least 3. Since L(ℝ) intersects each of the remaining d − 4 pseudo-lines, the degree of L · C is at least 3 + 3 + (d − 4) = d + 2. Contradiction.

In case 2, the degree of L(ℝ) · B is at least 5. Since L(ℝ) intersects each of the remaining d − 3 pseudo-lines, the degree of L · C is at least 5 + (d − 3) = d + 2. Contradiction.

In case 3, L(ℝ) is tangent to a real branch O of C that is not a pseudo-line, i.e., the degree of L(ℝ) · O is at least 2. Since L(ℝ) intersects each of the d − 2 pseudo-lines of C and since the degree of L · C is equal to d = 2 + (d − 2), the real line L(ℝ) is not tangent to a pseudo-line of C and is tangent to O at only one point. Contradiction, since L was supposed to be tangent to C at two distinct real points.

Let us, for completeness, include a proof of the following statement:

Lemma 5. Any pseudo-line of C contains at least 1 inflection point.

Proof. We show first the more general statement that, for any geometrically integral real plane curve D, the dual curve B^{⋎} of a smooth real branch B of D is homologically trivial in ℙ^{2}(ℝ)^{⋎}. Indeed, choose a general point P ∈ ℙ^{2}(ℝ) such that P ∉ B. Let p: B → ℙ^{1}(ℝ) be the restriction of the linear projection from ℙ^{2}(ℝ) \ {P } onto ℙ^{1}(ℝ) with center P. Then, p is a nonconstant real analytic map from the real analytic curve B into the real analytic curve ℙ^{1}(ℝ). Such a map is necessarily ramified at an even number of points of B. Therefore, the number of lines passing through P and tangent to B is even. Dually, this means that the line dual to P intersects B^{⋎} in an even number of points. Therefore, B^{⋎} is homologically trivial in ℙ^{2}(ℝ)^{⋎}.

Now, let B be a pseudo-line of C. It is clear that B^{⋎} is a real branch of C^{⋎}. By the preceding paragraph, B^{⋎} is not a smooth real branch of C^{⋎} since otherwise B = (B^{⋎})^{⋎} would be homologically trivial. Therefore, B^{⋎} contains singularities. By Proposition 4 (3), B^{⋎} can only contain ordinary cusps as singularities. Hence, B^{⋎} contains at least 1 ordinary cusp. It follows that B contains at least 1 inflection point.

Proof of Theorem 1. Let B be a pseudo-line of C. By Proposition 4 (2), B is a smooth real analytic curve in ℙ^{2}(ℝ). Let Q be a point on B and let L ⊆ ℙ^{2}(ℝ) be the tangent line at Q to B. Since B is a pseudo-line, L has to intersect B in yet another point P. Using Bezout’s Theorem, one sees that the point P ∈ B is uniquely determined by Q. Let α: B → B be the map defined by α(Q) = P. It is clear that α is continuous.

An inflection point of B is a fixed point of α and conversely. Therefore, we have to show that α has exactly 3 fixed points. The idea is to show that α is a topological covering of B of degree −2. It will then follow from Lemma 7 below that α has exactly 3 fixed points.

For P ∈ B, let π_{P} : ℙ^{2} \ {P } → ℙ_{P}^{1} be the linear projection with center P. Here, ℙ_{P}^{1} denotes the real algebraic curve of projective lines in ℙ^{2} passing through P. Of course, ℙ_{P}^{1} is isomorphic to ℙ^{1}, however, not canonically. That will be crucial below. By definition, the π_{P} -image of a real point Q of ℙ^{2} \ {P } is the real projective line passing through P and Q.

Let f_{P} : C̃ → ℙ_{P}^{1} be the unique morphism such that f_{P} = π_{P} ∘ ν on C̃ \ ν^{−1}(P). Then, f_{P} is a morphism of degree d − μ, where μ is the multiplicity of C at P or, equivalently, μ is the number of real branches of C passing through P. By Proposition 4 (2), one may identify a real branch of C with the corresponding real branch of C̃. The morphism f_{P} is ramified at a point Q ∈ B if and only if the real line L ⊆ ℙ^{2}(ℝ) through P and Q is tangent to B at Q. It follows that the fiber α^{−1}(P) is equal to the set of ramification points of the restriction of f_{P} to B.

Let B′ be a pseudo-line of C not passing through P. Since B′ is contained in ℙ^{2}(ℝ)\{P }, the restriction of f_{P} to B′ is a continuous map from B′ into ℙ^{1}(ℝ) of odd topological degree (see [7] for the notion of topological degree mod 2). In particular, the restriction of f_{P} to B′ is surjective. Since there are (d−2)−μ pseudo-lines of C not passing through P and since the degree of f_{P} is equal to d−μ, each fiber of the restriction of f_{P} to B has cardinality at most 2. In fact, more precisely, for all points R ∈ ℙ_{P}^{1} (ℝ), the degree of the divisor (f_{P} |_{B})^{⋆}(R) on B is at most 2. Since the topological degree of the restriction of f_{P} to B is even, there are either 0 or 2 points of B at which f_{P} is ramified. In the former case f_{P} |_{B} is not null-homotopic, in the latter case f_{P} |_{B} is null-homotopic.

Let T = T_{B/ℙ^{2}(ℝ)} be the restriction to B of the tangent bundle of ℙ^{2}(ℝ). Since B is a pseudo-line, the real analytic vector bundle T is isomorphic to the direct sum of a trivial line bundle and a nontrivial, i.e. a Möbius line bundle on B. Denote by ℙ(T ) the projectivization of T . The total space of ℙ(T ) is a Klein bottle. The fiber ℙ(T )_{P} of ℙ(T ) over a point P of B is canonically isomorphic to ℙ_{P}^{1} (ℝ), i.e., we have made the collection of all real projective lines {ℙ_{P}^{1} (ℝ)}_{P ∈B} into a locally trivial real analytic fiber bundle over B. Define

F: B × B ⟶ ℙ(T )

by F (P, Q) = f_{P} (Q) for all (P, Q) ∈ B × B. The map F is real analytic, and, when we consider B × B to be fibered over B through the projection on the first factor, F is a map of locally trivial real analytic fiber bundles over B. The fiber F_{P} of F over P ∈ B is the map f_{P} |_{B}, i.e., we have made the collection of all maps f_{P} |_{B} into a real analytic family of maps over B.

Above, we have seen that f_{P} |_{B} is ramified at exactly 0 or 2 points of B. Moreover, f_{P} |_{B} is not null homotopic in the former case and is null homotopic in the latter case. Since the maps f_{P} |_{B} vary continuously in a connected family, either all maps f_{P} |_{B} are null homotopic, or all maps f_{P} |_{B} are not null homotopic. Hence, either all maps f_{P} |_{B} are unramified, or all maps f_{P} |_{B} are ramified at exactly 2 points. Now, there is, of course, a point P_{0} ∈ B such that α^{−1}(P_{0}) is nonempty. But then, as we have seen above, f_{P_{0}}|_{B} is ramified. Hence, all maps f_{P} |_{B} are ramified at exactly 2 points of B. Moreover, since, for all R ∈ ℙ_{P}^{1} (ℝ), the degree of the divisor (f_{P} |_{B})^{⋆}(R) is at most 2, the image (f_{P} |_{B})(B) is an interval I_{P} in ℙ_{P}^{1} (ℝ). The union I of all I_{P} is an interval subbundle of ℙ(T ). Since the latter fiber bundle is not globally trivial, the interval bundle I is a Möbius bundle over B. This implies that the ramification locus of F, i.e., the union of the ramification loci of the maps f_{P} |_{B}, is a nontrivial topological covering of the base B of degree ±2. Now, this ramification locus is the transpose of the graph of α. Hence, α is of degree ±2.

In order to show that α is of degree −2, recall that B has at least 1 inflection point by Lemma 5 and that such an inflection point is necessarily ordinary by Proposition 4 (3). A local study of α at an ordinary inflection point of B reveals that α is orientation-reversing. Hence, the topological degree of α is equal to −2. It follows from Lemma 7 below that the number of fixed points of α is equal to 3. Therefore, B contains exactly 3 inflection points.

Before proving Lemma 7 one needs the following preliminary statement:

Lemma 6. Let α, β: S^{1} → S^{1} be topological coverings either both orientation-preserving or both orientation-reversing. Then, the product αβ: S^{1} → S^{1}, defined by (αβ)(z) = α(z) · β(z) for any z ∈ S^{1}, is also a topological covering.

Proof. Let p: ℝ → S^{1} be the universal covering defined by p(t) = exp(2πit) for t ∈ ℝ. Let α̃, β̃: ℝ → ℝ be liftings of α and β, respectively, i.e., p ∘ α̃ = α ∘ p and p ∘ β̃ = β ∘ p. Then, α̃+β̃ is a lifting of αβ. Since α and β are topological coverings, α̃ and β̃ are homeomorphisms of ℝ onto itself. Since α and β are either both orientation-preserving or both orientationreversing, α̃ and β̃ are either both strictly ascending or both strictly descending real functions. It follows that α̃ + β̃ is strictly ascending or strictly descending. In particular, α̃ + β̃ is a homeomorphism and, therefore, αβ is a topological covering.

Lemma 7. Let α: S^{1} → S^{1} be a topological covering of degree −e, for some e > 0. Then, α has exactly e + 1 fixed points.

Proof. Let β: S^{1} → S^{1} be the standard topological covering of degree −e, i.e., β(z) = z̄^{e} for z ∈ S^{1}. Then, α and β are isotopic coverings, i.e., there is a homotopy F: S^{1} × [0, 1] → S^{1} such that F_{0} = α, F_{1} = β and F_{t} is a topological covering for all t ∈ [0, 1]. Define F′: S^{1} × [0, 1] → S^{1} by F′(z, t) = F (z, t) · z̄ for (z, t) ∈ S^{1} × [0, 1]. By Lemma 6, F′ is an isotopy of topological coverings of S^{1}. One has F′_{1}(z) = z̄^{e+1}. Hence, the fiber F′_{1}^{−1}(1) consists of e + 1 points. Then, the fiber F′_{0}^{−1}(1) consists of e + 1 points too, i.e., there are exactly e + 1 points z ∈ S^{1} such that α(z)z̄ = 1. Therefore, α has exactly e + 1 fixed points.

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Received March 1, 2000