Homogeneous Finite Projective Plane

• A homogeneous finite projective plane is a finite geometry where points and lines follow strict incidence axioms and display maximal, transitive symmetry via automorphism groups.
• Group-theoretic constructions and planar functions are key methods for building these planes, providing concrete algebraic frameworks and coset representations.
• These structures underpin applications in design theory, coding theory, and algebraic curves, with ongoing research exploring their universality, classification, and symmetry properties.

A homogeneous finite projective plane is a combinatorial and geometric structure in which the incidence relations between points and lines exhibit a maximal degree of symmetry, typically ensured by the presence of a transitive collineation (automorphism) group. The paper of such planes synthesizes deep algebraic, combinatorial, and geometric principles, especially as these structures connect to groups, rings, incidence complexes, and design theory.

1. Fundamental Definition and Characterization

A finite projective plane of order $n$ is a set of points and lines satisfying the axioms:

• Each line contains exactly $n+1$ points,
• Each point lies on exactly $n+1$ lines,
• Any two distinct points determine a unique line,
• Any two distinct lines intersect in precisely one point.

The plane is called homogeneous if its automorphism group acts transitively (or at least highly transitively) on the relevant combinatorial structures—typically either on points, lines, flags, or more generally, the set of flags $(x, L)$ with $x$ not incident to a distinguished line and $L$ not incident to a distinguished point (Kantor, 21 Aug 2024). This strong symmetry implies that "local" geometric and algebraic characteristics extend globally across the plane.

The smallest examples include the Fano plane ($n=2$) and order-$3$ plane ($n=3$), both of which are strictly homogeneous in the ultrahomogeneity sense (Kubiś et al., 2022).

2. Group-Theoretic and Algebraic Constructions

A central paradigm for constructing homogeneous finite projective planes uses groups and their coset structures. In (Kantor, 21 Aug 2024), a projective plane $T$ can be built from a group $G$ with subgroups $A$, $B$, and $M$ subject to conditions:

• $|A| = |B| = |M| = n\cdot k$ and $|G| = n^3k$,
• $AM$ and $BM$ are subgroups of order $n^2k$,
• $G = AMB$,
• For $a \in A \setminus(A \cap B)$, $b \in B \setminus(A \cap B)$, $ab \neq b'a'$ for all $b' \in B$, $a' \in A$.

The resulting incidence geometry emerges by taking cosets as points or lines and declaring incidence by nonempty intersection of cosets. These criteria generalize classical translation planes (where $G$ is abelian and $A$, $B$ generate the translation group), as well as semifield planes and other known cases. In all known examples, $n$ is a prime power due to the inherent structural constraints, though this method potentially admits new configurations if suitable groups can be constructed (Kantor, 21 Aug 2024).

A homogeneous plane's collineation group may fix a distinguished flag $(\infty, L_{\infty})$ and act transitively on all flags not incident to the fixed point or line, which encodes significant symmetry and coordinateability.

3. Algebraic and Combinatorial Structures: Incidence Rings, Nets, and Functions

3-Nets and Group Realizations

A dual 3-net consists of three disjoint sets (components) of points within a projective plane, with the property that any line meeting two components meets all three, and a binary operation is naturally encoded via collinearity (Korchmaros et al., 2011). The classification of 3-nets reveals several infinite families:

• Proper algebraic dual 3-nets: Points constitute cosets on (possibly singular) cubic curves, and collinearity is given via the group law, such as $g_1+g_2+g_3=0$.
• Triangular and conic-line dual 3-nets: Points lie on triangles and conics respectively, often realizing cyclic groups.
• Tetrahedron-type nets: Realize dihedral groups, with a decomposition of components into half-sets.
• Sporadic examples: E.g., Urzúa's net realizing the quaternion group of order $8$.

This comprehensive classification confirms that every finite group realizable as a group of collineations in the plane appears via some 3-net configuration, linking the algebraic group structure with geometric symmetry (Korchmaros et al., 2011).

Planar Functions

Planar functions $f: \mathbb{F}_q \rightarrow \mathbb{F}_q$ are used to construct finite projective planes, with properties tailored to the field characteristic. In characteristic $2$, the definition is modified (Zhou's definition: $c \mapsto F(c+d)+F(c)+cd$ is bijective) to maintain the necessary permutation properties (Scherr et al., 2013).

Homogeneous planar functions satisfy $f(\lambda x) = \lambda^d f(x)$ for all $\lambda \in \mathbb{F}_p$, with $d \equiv 2 \pmod{p-1}$; over $\mathbb{F}_{p^2}$, $x^2$ is the unique case up to equivalence (Feng, 2013). These functions are deeply linked to the construction of ovals and hence conics in finite projective planes.

Incidence Complexes and Associated Algebras

Finite projective planes naturally yield incidence complexes; one can associate:

• Stanley-Reisner ring $R/I_\Lambda$: Encodes combinatorial geometry via forbidden non-faces (e.g., noncollinear triples as generators).
• Inverse system algebra $R/I_\Delta$: Arises from Macaulay's inverse systems, relying on square-free monomials (lines) and their annihilators.

Graded Betti numbers and Lefschetz properties reflect deep combinatorial and algebraic constraints. For the inverse system algebra, the Weak/Strong Lefschetz Properties hold if char $K$ is $0$ or exceeds $q+1$ (for plane order $q$) (II et al., 2015).

4. Symmetry, Homogeneity, and Universality

Homogeneous projective planes feature highly transitive automorphism groups. Notably, some planes (e.g., the Fano plane and order-$3$ plane) are strictly homogeneous: every isomorphism between finite subplanes extends to a global automorphism (Kubiś et al., 2022). Under continuum hypothesis (CH), a universal projective plane of cardinality $\aleph_1$ exists, homogeneous with respect to its countable and finite subplanes.

Recent work on universal homogeneous matroids of rank 3 and their free projective extensions shows that every finite simple rank-3 matroid embeds in a single countable projective plane whose automorphism group contains the full infinite symmetric group $\mathrm{Sym}(\omega)$ (Paolini, 2017). This underscores a model-theoretic form of homogeneity.

Extensions and constructions (such as those using rings of lower triangular matrices) demonstrate that entire families of affine or projective planes can be built on shared point sets, further emphasizing universality and homogeneity when incidence is encoded via algebraic modules (Bartnicka et al., 2019).

5. Constraints, Non-Existence, and Classification

Rigorous combinatorial and algebraic methods constrain the existence and structure of finite projective planes:

• Coding-theoretic approaches: Non-existence of order-$10$ planes is shown using binary codes derived from the incidence matrix of the plane and analysis of weight enumerators (Perrott, 2016, Matolcsi et al., 2018).
• Linear programming and Delsarte LP-bound: Using Fourier analytic and LP-based bounds, one can rigorously prove non-existence for certain orders (e.g., $n=6$), and uniqueness for $n=7$ (Matolcsi et al., 2018).
• Group and collineation symmetries: The group-theoretic construction provides a systematic classification, covering all known planes admitting collineation groups fixing a flag and being flag-transitive; all constructed examples to date are of prime power order (Kantor, 21 Aug 2024).

The embedding problem and amalgamation properties establish necessary and sufficient conditions for universality and further restrict possible structures (Kubiś et al., 2022).

6. Algebraic Curves and Geometric Extremals

In Galois planes $\mathrm{PG}(2, q)$, Tallini curves are irreducible curves of degree $q+2$ containing all points of the plane (Cunha, 2018). These curves exhibit rich automorphism groups (e.g., Singer cycles of order $q^2+q+1$), encode deep arithmetic via the Weierstrass semigroup and Hasse–Witt invariants, and represent extremal cases for algebraic intersections in finite geometry.

Quotient curves under group action illuminate symmetries and connections to other well-known curves, while the ordinarity property (having maximal possible $p$-torsion points in the Jacobian) anchors the curve’s arithmetic relevance.

7. Directions for Research and Generalizations

Key open problems and directions include:

• Extending group-theoretic constructions to non-prime-power orders, especially by discovering suitable groups fulfilling all combinatorial and algebraic constraints (Kantor, 21 Aug 2024).
• Classifying planar and homogeneous planar functions in higher degree extensions and characteristics (Feng, 2013).
• Generalizing the mutual orthogonality (orthogoval) of projective and affine spaces to higher dimensions, where lines or subspaces of one become caps or arcs in the other (Saaltink, 19 Mar 2024).
• Investigating the connection between amalgamation properties (weak amalgamation property, Conjecture (t)) and universality/homogeneity (Kubiś et al., 2022).
• Exploring the linkage between ring-theoretic geometry and projective plane construction, with extension to broader classes of rings and module structures (Bartnicka et al., 2019).

These themes reflect both the strict algebraic-combinatorial foundation of homogeneous finite projective planes and the dynamism of modern research exploring universality, symmetry, and new possible avenues for construction and classification.