Self-intersections of surfaces that contain two circles through each point

Technical Summary: Self-intersections of surfaces that contain two circles through each point

Problem Statement
The paper addresses a fundamental problem in real algebraic geometry and Möbius geometry: characterizing the self-intersection loci of surfaces in $\mathbb{R}^3$ (and their counterparts in the unit 3-sphere $S^3$) that contain at least two circles passing through almost every point. While surfaces containing two lines through each point are well-understood as doubly ruled quadrics (e.g., the hyperboloid of one sheet) with no self-intersections, the analogous problem for circles is significantly more complex. The author seeks to classify the topology and geometry of the singular loci (self-intersections) of such "celestial surfaces," specifically those of degree 8, and to understand how the circles comprising the surface interact with these singularities.

Methodology
The investigation employs a synthesis of Möbius geometry, algebraic geometry, and topology. The core methodological framework involves:

1. Möbius Equivalence and Models: The study is conducted on the Möbius quadric $S^3 \subset \mathbb{P}^4$. Surfaces are analyzed via their Euclidean models (via stereographic projection) and spherical models (via the identification of $S^3$ with unit quaternions). The paper distinguishes between "Bohemian" surfaces (pointwise sums of circles in $\mathbb{R}^3$) and "Cliffordian" surfaces (pointwise products of circles in $S^3$, related to Clifford translations).
2. Algebraic Geometry of Del Pezzo Surfaces: Celestial surfaces of degree 8 are identified as linear projections of del Pezzo surfaces of degree 8. The author utilizes the Néron-Severi lattice $N(X)$ of the smooth model $O(X) \cong \mathbb{P}^1 \times \mathbb{P}^1$ to classify divisor classes. The circles on the surface correspond to specific classes ($\ell_0, \ell_1$), and singular components are analyzed via their bidegrees and intersection properties within this lattice.
3. Stereographic Projections and Singularities: The paper analyzes complex linear projections of the surface $X$ from points $p$ in its singular locus to $\mathbb{P}^3$. By studying the resulting projected surfaces (which are also del Pezzo surfaces of lower degree) and their "cyclicity" (multiplicity of the absolute conic), the author derives constraints on the number of concurrent complex lines passing through singular points.
4. Delta Invariants and Topological Classification: The "sectional delta invariant" is used to relate the arithmetic and geometric genera of the surface to its singular components. This allows for the classification of the "pseudo singular types" (incidence relations of singular components) and the subsequent determination of the "visible types" (the topology of the real singular locus).
5. Computational Verification: Explicit parametrizations of celestial surfaces are generated using rational maps defined by parameters $(\tau_i, \mu_i)$. Gröbner basis computations are employed to determine the singular types and visible types of specific examples, verifying the theoretical classifications.

Key Contributions and Results

• Classification of Singular Types (Theorem I): The paper proves that for a celestial surface $X \subset S^3$ of degree 8, the singular type is restricted to one of five classes: $D_1, D_2, D_3, D_4,$ or $D_5$. These types are defined by the incidence relations of complex double curves (e.g., Viviani curves, twisted quartics, conics, or pairs of circles) and their multiplicities.
  • $D_1$ and $D_2$ correspond to Bohemian surfaces.
  • $D_3, D_4,$ and $D_5$ correspond to Cliffordian surfaces.
• Topological Classification of Visible Loci (Corollary III): The paper classifies the homeomorphism types of the visible singular locus (the real self-intersection). The possible topological types are represented by symbols: $=$, $+$, $\circ$, $-$, $\infty$, $\alpha$, and $\sigma$. For instance, a surface of type $D_4$ may have a visible locus homeomorphic to an infinity symbol ($\infty$), a cross ($+$), or an arc ($\alpha$).
• Relationship between Geometry and Topology: The author characterizes how a circle in the surface meets the singular locus as it moves in its pencil. For example, in a surface of type $D_4[\infty]$, a generic circle meets the double circle of the singularity in two points, whereas in $D_4[\alpha]$, it may meet tangentially in one point.
• Explicit Parametrizations: The paper provides a "parametric type" tuple $(\tau_0, \dots, \tau_3; \mu_0, \dots, \mu_3)$ that encodes explicit rational parametrizations for celestial surfaces of degree 8. Tables 2, 3, and 4 list specific examples mapping these parameters to the classified singular and visible types.
• Algorithmic Identification: The paper presents algorithms (Remarks 45–47) to compute the singular and visible types of a surface given two generating circles $A$ and $B$ (for Cliffordian surfaces) or two circles in $\mathbb{R}^3$ (for Bohemian surfaces), utilizing elimination ideals and Gröbner bases.

Significance and Claims
The paper claims to provide a complete classification of the singular loci for real celestial surfaces of degree 8, a class of surfaces that generalizes the well-known Darboux cyclides (degree 4). The significance lies in:

1. Refining Previous Work: It refines the classification of celestial surfaces by Skopenkov and Krasauskas [29] by distinguishing between Bohemian and Cliffordian types based on their singular structures.
2. Topological Insight: It offers a detailed understanding of the topology of self-intersections, moving beyond isolated singularities to classify curves of self-intersection and their interaction with the surface's generating circles.
3. Unification of Geometries: The work unifies Euclidean, spherical, and Möbius geometries by showing how properties of surfaces in $\mathbb{R}^3$ translate to properties of surfaces in $S^3$ and their algebraic models in $\mathbb{P}^4$.
4. Computational Utility: By providing explicit parametrizations and algorithms, the paper enables the construction and visualization of these complex surfaces, facilitating further study in geometric modeling and kinematics.

The author notes that while the classification of singular types is proven, the classification of visible types for certain cases (specifically $D_4[\sigma]$, $D_4[=]$, and $D_4[-]$) remains a conjecture (Conjecture 2), suggesting that these topological configurations may not be realizable by real celestial surfaces.