Real Algebraic Curves with Involution

• Real algebraic curves with involution are curves over ℝ endowed with an involutive map (typically antiholomorphic) that distinguishes their real loci through fixed-point sets.
• They link real algebraic geometry with Riemann surface theory, moduli spaces, and integrable systems via explicit computations such as period matrices and Smith normal forms.
• The classification based on topological invariants and automorphism groups offers refined insights into their spectral properties and computational tractability.

A real algebraic curve with involution is an algebraic curve $C$ over $\mathbb{R}$ equipped with an involutive morphism—typically antiholomorphic, sometimes holomorphic—that endows $C$ with distinguished geometric, topological, and group-theoretic structures. Antiholomorphic involutions model complex conjugation and define a fixed-point real locus, providing a bridge between real algebraic geometry, Riemann surface theory, moduli spaces, and representation theory. Such curves underpin the equivariant paper of complex structures, the classification of real varieties, birational geometry, and arise as spectral data in the theory of invariant hypercomplex structures and integrable systems.

1. Definitions and Types of Involutions

Let $C$ be a nonsingular algebraic curve of genus $g$ over $\mathbb{R}$, considered as a compact Riemann surface $S$. An antiholomorphic involution $\tau:C\to C$ satisfies $\tau^2=\id$ and acts on affine coordinates $(x, y)$ via complex conjugation: $\tau(x, y) = (\overline{x}, \overline{y}).$ Its fixed point set $C(\mathbb{R})$—the real points—decomposes into $k$ disjoint simple closed curves, or "real ovals". For holomorphic involutions $\sigma:C\to C$ with $\sigma^2=\id$, one typically requires that $\tau$ and $\sigma$ commute, $\sigma\circ\tau = \tau\circ\sigma$ (Sheinman, 6 Nov 2025).

Topological type is encoded as $(g, k, \epsilon)$, where $g$ is the genus, $k$ the number of real ovals, and $\epsilon=1$ for separating (dividing) curves where $C\setminus C(\mathbb{R})$ is disconnected, $\epsilon=0$ otherwise (Kalla et al., 2012, Sheinman, 6 Nov 2025). Harnack's theorem states $0\le k\le g+1$, with equality indicating an $M$-curve.

2. Correspondence with Symmetric Geometric Structures

A central development is the bijection between $U(k)$-invariant hypercomplex structures on regular semisimple adjoint orbits in $\mathfrak{gl}(k, \mathbb{C})$, and triples $(C, \pi, \sigma)$ where $C$ is a real algebraic curve of genus $g=(k-1)^2$, $\pi:C\to\mathbb{P}^1$ is a flat degree-$k$ projection, and $\sigma:C\to C$ is an antiholomorphic involution covering the antipodal map $\zeta\mapsto -1/\overline{\zeta}$ on $\mathbb{P}^1$ (Bielawski, 2021). Explicitly,

$$\sigma(\zeta, \eta) = \left(-1/\overline{\zeta}, -\overline{\eta}/\overline{\zeta}^2\right),$$

and $C$ is cut out by a characteristic polynomial equation in the total space of $\mathcal{O}_{\mathbb{P}^1}(2)$: $\det(\eta - A(\zeta)) = 0$ for a quadratic $A(\zeta)$ in $\zeta$ with coefficients in $\mathfrak{u}(k)$. The moduli space of such curves $(C, \pi, \sigma)$ has real dimension $2k^2-2k$.

3. Computation of Topological Type and Homological Invariants

Given a real plane algebraic curve $R: f(x, y)=0$ with coefficients in $\mathbb{R}$, the antiholomorphic involution $\tau$ acts on homology and differentials. One introduces a canonical symplectic basis $\{\mathcal{A}_i, \mathcal{B}_i\}_{i=1}^g$ with intersection pairings $\mathcal{A}_i\cdot\mathcal{B}_j=\delta_{ij}$. The involution's action on homology is encoded by an integer matrix $R$ diagonalizable over $\mathbb{Q}$ (Kalla et al., 2012).

An algorithm, based on symplectic transformations and Smith normal forms, computes a basis where $\mathcal{A}$-cycles are fixed by $\tau$ and deduces the topological invariants $(g, k, a)$. The matrix $H$ (over $\mathbb{Z}/2\mathbb{Z}$), extracted from these computations, determines $k$ and the dividing/non-dividing character $a$ via its rank and block structure.

Example Output Table

Curve	Genus $g$	Ovals $k$	Dividing $a$
Trott (quartic)	3	4	0 (dividing)
Klein's curve ($y^7-\cdots$)	3	1	1 (non-div.)
Fermat quartic	3	0	1 (non-div.)
Fermat quintic	6	1	1 (non-div.)

These numerical procedures are computationally tractable for genus $g\le 6$ using existing symbolic or numerical algebra systems.

4. Prym Varieties, Abel–Prym Maps, and Real Spectral Data

When a real curve admits a holomorphic involution $\sigma$ commuting with the antiholomorphic $\tau$, the associated Prym variety and Abel–Prym map carry additional real structure. The Prym variety is defined using the anti-invariant differentials under $\sigma$, with period matrix $\Pi$ constructed as

$\Pi_{ij} = \int_{b_j}\tilde{\omega}_i$

for anti-invariant holomorphic 1-forms $\tilde{\omega}_i$. The isoPrymian $$\mathrm{isoPrym}(X,\sigma)=\mathbb{C}^h/(2\pi i\,\mathbb{Z}^h \oplus \Pi\,\mathbb{Z}^h)$$ is an unramified cover of the Prym variety (Sheinman, 6 Nov 2025).

The Abel–Prym map $A:\mathrm{Div}^0(X)\to\mathrm{isoPrym}(X,\sigma)$ is inverted by the Riemann–Prym vanishing theorem: divisors corresponding to zeros of the Prym theta function $\theta(z, \Pi)$ are parametrized by explicit formulas involving period shifts and symmetric functions of local coordinates. Realness (Novikov–Veselov conditions) for spectral data is encoded in constraints on $z$,

$z + t\overline{z} = \Delta + t\overline{\Delta}$

or its variants, determining whether the inverse image is $\tau$- or $(\sigma\tau)$-invariant.

Example: Real Elliptic Curve

For $X=\{y^2=x(x-1)(x-\lambda)\}$, with $\lambda\in(0,1)$, the real locus consists of one oval, and the Abel–Prym inversion is realized via classical elliptic function theory.

5. Birational Involutions and Classification in the Real Projective Plane

Birational involutions $\alpha:\mathbb{P}^2\to\mathbb{P}^2$ over $\mathbb{R}$, defined by degree $d$ homogeneous polynomials satisfying $\alpha^2=\id$, are classified according to their behavior on fixed curves (Mangolte, 23 May 2025). Over $\mathbb{R}$, there are twelve conjugacy classes; some fix no irrational curve, others fix curves of geometric genus $g\geq 1$, including:

• Bertini: $g=4$, non-hyperelliptic
• Geiser: $g=3$, non-hyperelliptic
• Kowalevskaya: $g=1$, elliptic
• de Jonquières: $g\ge 1$, hyperelliptic
• Twisted Iskovskikh: $g=d-1$

Unlike the complex case, the curve $C$ alone fails to determine the conjugacy class of $\alpha$ in $\mathrm{Bir}_\mathbb{R}(\mathbb{P}^2)$. For real hyperelliptic curves of genus $\ge2$ with at least two ovals, there exist uncountably many non-conjugate involutions fixing an isomorphic $C$.

6. Moduli Spaces and Dimension Counts

The moduli space of real algebraic curves with antiholomorphic involution (with or without boundary/punctures) decomposes into connected components indexed by the topological type $t=(g,k,\epsilon\,|\, 2n_1,\,2m_1,\,n_R,\,m_R)$. Each component is homeomorphic to a quotient $V_t/\Delta_t$, where $V_t$ is a real vector space of explicit dimension (Natanzon et al., 2020): $\dim T_t = 3g - 3 + 3n_1 + 2m_1 + 2n_R + m_R$ In the compact (no boundary/punctures) case, $\dim V = 3g-3$ matches the known dimension for real Teichmüller spaces.

7. Hyperelliptic Curves of Genus 3 and Involution

Genus 3 hyperelliptic curves admitting an extra involution are classified and parametrized by three dihedral invariants $(u_1, u_2, u_3)$ (Gutierrez et al., 2012). The normal form

$Y^2 = x^8 + a_3x^6 + a_2x^4 + a_1x^2 + 1$

with involution $\sigma(x, y) = (-x, y)$ is birationally equivalent to a locus $L_3$ in the moduli space $H_3$, with stratification by automorphism group given by relations among $(u_1, u_2, u_3)$:

• $V_4$ generic, higher symmetry loci for special relations.

A key result is that for $|\mathrm{Aut}(X)|>2$, the field of moduli is a field of definition. When $|\mathrm{Aut}(X)|>4$, explicit rational models over the field of moduli are constructed.

Summary

Real algebraic curves equipped with involution exhibit a rich interplay between topological invariants $(g, k, \epsilon)$, moduli spaces, equivariant deformation theory, and representation-theoretic phenomena. Classification schemes, both birational and automorphism-theoretic, reveal subtleties unique to the real case, including non-rigidity and spectral degeneracy. Algorithmic techniques using period matrices, Smith normal forms, and symmetric invariants enable computational exploration of these curves' structure and moduli, with applications spanning real algebraic geometry, theta function theory, integrable systems, and equivariant complex geometry.