Plane rectifiable curves: old and new

Technical Summary: Plane Rectifiable Curves: Old and New

Problem Statement
The paper addresses the classical problem of algebraic rectifiability for plane algebraic curves. A curve is defined as algebraically rectifiable if its arc length, viewed as a function of the endpoints, is an algebraic function. While historically significant examples exist (such as the semicubic parabola), the general criteria for determining when an arbitrary algebraic curve possesses this property have been framed primarily through geometric constructs like evolutes. The authors seek to recast this problem using the language of meromorphic differentials on the normalization of algebraic curves, aiming to provide intrinsic criteria and unify the theory with the study of strata of differentials.

Methodology
The authors employ the theory of meromorphic differentials on compact Riemann surfaces. Their primary methodological shift involves associating an algebraic plane curve $\Gamma$ with a specific quadratic differential $ds^2 = dx^2 + dy^2$ defined on its normalization $\tilde{\Gamma}$.

1. Differential Formulation: They define the arc length 1-form $ds$ and the associated quadratic differential $ds^2$ on the normalization. They analyze the divisor of zeros and poles of this differential, relating them to isotropic tangencies (tangents passing through circular points at infinity) and the behavior at infinity.
2. Realization Criteria: The paper investigates which quadratic differentials on a real Riemann surface $(X, \rho)$ can be realized as the arc length differential of a real meromorphic plane map. This involves splitting the quadratic differential $q$ into conjugate 1-forms $q = \omega \rho^*\omega$ and imposing exactness conditions (vanishing periods) on $\omega$.
3. Higher Order Generalization: The framework is extended to $k$-differentials. The authors define "exact" $k$-differentials via the exactness of the canonical $k$-cover and relate this to the algebraicity of Abelian integrals $\int \phi^{1/k}$.
4. Affine Analogue: The methodology is adapted to special affine geometry, replacing the Euclidean quadratic differential with an affine cubic differential $(\gamma' \wedge \gamma'')dt^3$.

Key Contributions and Results

• Intrinsic Realization Criterion (Theorem 4): The authors establish that a $\rho$-real quadratic differential $q$ on a compact real Riemann surface $X$ is induced by a real meromorphic plane map if and only if $q$ admits an exact admissible half-divisor. This means $q$ must split as $\omega \rho^*\omega$ where $\omega$ is an exact meromorphic 1-form (i.e., $\omega = dz$ for a meromorphic function $z$). This separates the existence problem into a divisor-theoretic condition and a period condition.
• Uniqueness (Proposition 5): For a fixed splitting (divisor), the realization is unique up to direct Euclidean motion. If the differential is scaled by a positive constant, the realization is unique up to Euclidean similarity.
• Equivalence of Rectifiability Conditions (Theorem 12): The paper proves that for a non-linear plane algebraic curve, the following are equivalent:
  1. The curve is algebraically rectifiable.
  2. The canonical double cover of the arc length quadratic differential is exact.
  3. The involutes of the curve are algebraic.
  4. The curve is the evolute of an algebraic curve.
     This result reinterprets Humbert's classical theorem (which states a curve is rectifiable iff it is an evolute) as the geometric manifestation of the exactness criterion for the arc length differential.
• Classification of Rational Curves (Theorem 13): For non-linear rational real plane curves, if the arc length quadratic differential has exactly two singularities, the curve must be a circle (up to Euclidean similarity and reparametrization). If the singularities are real, the curve is a straight line.
• Exactness of Higher Order Differentials (Theorem 7): The authors prove that for a meromorphic $k$-differential $\phi$, the exactness of its canonical $k$-cover is equivalent to the algebraicity of the Abelian integral $\int \phi^{1/k}$ and to the representation $\phi = f (df/f)^k$ for some meromorphic function $f$.
• Affine Rectifiability (Proposition 15): An analogous result is provided for affine geometry: affine involutes are algebraic if and only if the affine cubic differential has an exact canonical cover.

Examples and Computations
The paper computes and classifies the arc length differentials for several specific curves, locating them within the appropriate strata of meromorphic differentials ($\Omega^2 M_g$):

• Conics: Circles are the only rectifiable conics. Hyperbolas and non-circular ellipses yield differentials with four simple zeros and two poles of order four. Parabolas yield differentials with two simple zeros and one pole of order six.
• Bernoulli Lemniscate: The arc length differential has four simple poles and no zeros, corresponding to a "pillowcase" flat surface.
• Semicubic Parabola: A classical rectifiable example where the differential is exact, belonging to a specific stratum with a zero of order two and a pole of order eight.
• Pythagorean-Hodograph Curves: The paper discusses a quintic Pythagorean-hodograph curve, demonstrating a smooth, rectifiable curve arising from a non-linear identity, distinct from the singular semicubic parabola.

Significance
The paper claims to provide a "useful bridge" between classical projective geometry and the modern theory of strata of meromorphic differentials. By recasting algebraic rectifiability in the language of exact differentials, the authors offer a uniform framework that:

1. Clarifies the relationship between rectifiability, evolutes, and involutes.
2. Generalizes the concept to higher-order differentials and affine geometry.
3. Provides intrinsic, divisor-theoretic criteria for determining rectifiability, moving beyond purely geometric characterizations.

The work does not propose new experimental applications but rather refines the theoretical understanding of when the Euclidean (or affine) metric restricted to an algebraic curve becomes algebraically integrable. It connects historical results by Serret, Laguerre, and Humbert with modern developments in the moduli spaces of differentials and Pythagorean-hodograph curves.