[Paper Review] The tangent splash in $\PG(6,q)$

S. G. Barwick, Wen‐Ai Jackson|arXiv (Cornell University)|May 29, 2013

Finite Group Theory Research5 references4 citations

TL;DR

This paper investigates the geometric structure of tangent splashes—sets of $q^2+1$ points on the line at infinity—associated with order-$q$ subplanes in $\mathrm{PG}(2,q^3)$ via the Bruck-Bose representation in $\mathrm{PG}(6,q)$. It constructs the unique order-$q$ subplane from a given tangent splash and a fixed order-$q$ subline using ruled surfaces and transversal lines, proving uniqueness through projective equivalence in the cubic extension.

ABSTRACT

Let B be a subplane of PG(2,q^3) of order q that is tangent to $\ell_\infty$. Then the tangent splash of B is defined to be the set of q^2+1 points of $\ell_\infty$ that lie on a line of B. In the Bruck-Bose representation of PG(2,q^3) in PG(6,q), we investigate the interaction between the ruled surface corresponding to B and the planes corresponding to the tangent splash of B. We then give a geometric construction of the unique order-$q$-subplane determined by a given tangent splash and a fixed order-$q$-subline.

Motivation & Objective

• To characterize the interaction between ruled surfaces representing order-$q$ subplanes and the planes corresponding to their tangent splashes in $\mathrm{PG}(6,q)$.
• To provide a geometric construction of the unique order-$q$ subplane with a given tangent splash and a fixed order-$q$ subline in the Bruck-Bose representation.
• To establish the correspondence between the ruled surface in $\mathrm{PG}(6,q)$ and the tangent subplane via shared directrices and transversal lines.
• To prove that the constructed subplane is uniquely determined by the splash and subline, using projective equivalence in the cubic extension.

Proposed method

• Represent $\mathrm{PG}(2,q^3)$ via the Bruck-Bose construction in $\mathrm{PG}(6,q)$ using a regular 2-spread in $\mathrm{PG}(5,q)$.
• Map the order-$q$ subplane $\mathscr{B}$ to a ruled surface $\mathcal{V}$ in $\mathrm{PG}(6,q)$ with a twisted cubic and a conic as directrices.
• Correspond the tangent splash $\mathscr{S}_T$ to $q^2+1$ planes in $\mathrm{PG}(6,q)$, each intersecting the spread element $[T]$.
• Use the transversal lines $g$, $g^q$, $g^{q^2}$ of the regular spread to define a projectivity $\phi$ mapping points on the twisted cubic $\mathcal{N}$ to points on the conic $\mathcal{C}$.
• Construct the ruled surface $\mathcal{V}$ as the image of a projectivity $\eta$ mapping the twisted cubic $\mathcal{N}$ to the conic $\mathcal{C}$, preserving the transversals.
• Prove uniqueness by showing that the constructed ruled surface $\mathcal{V}$ and the surface $[\mathscr{B}]$ corresponding to the unique subplane $\mathscr{B}$ share the same directrices and transversals, hence are equal.

Experimental results

Research questions

• RQ1 How do the planes corresponding to a tangent splash interact with the ruled surface representing a tangent order-$q$ subplane in $\mathrm{PG}(6,q)$?
• RQ2 Can a geometric construction in $\mathrm{PG}(6,q)$ uniquely recover the order-$q$ subplane from a given tangent splash and a fixed order-$q$ subline?
• RQ3 What role do the transversal lines of the regular 2-spread play in characterizing the ruled surface of a tangent subplane?
• RQ4 How is the tangent space of a point on the ruled surface in $\mathrm{PG}(6,q)$ related to the tangent splash in $\mathrm{PG}(2,q^3)$?
• RQ5 What projective invariants ensure the uniqueness of the constructed subplane from the splash and subline?

Key findings

• The tangent splash $\mathscr{S}_T$ of a tangent order-$q$ subplane $\mathscr{B}$ corresponds to $q^2+1$ planes in $\mathrm{PG}(6,q)$, each intersecting the spread element $[T]$.
• The ruled surface $\mathcal{V}$ in $\mathrm{PG}(6,q)$ corresponding to $\mathscr{B}$ contains the twisted cubic $\mathcal{N}=[\ell]$ and a conic directrix, and is determined by a projectivity $\eta$ mapping $\mathcal{N}$ to the conic.
• The ruled surface $\mathcal{V}$ and the surface $[\mathscr{B}]$ are equal because they share the same conic directrix, twisted cubic directrix, and transversal lines of the regular spread.
• The constructed order-$q$ subplane in $\mathrm{PG}(2,q^3)$ is uniquely determined by the tangent splash $\mathscr{S}_T$ and the order-$q$ subline $\ell$, as proven by the uniqueness of the projectivity $\eta$.
• The projectivity $\phi$ mapping $N_e$ to $C_e$ is induced by a matrix transformation and extends to a projectivity in $\mathrm{PG}(6,q^3)$, ensuring the surface is well-defined.
• The proof confirms that the constructed subplane $\pi$ is identical to the unique subplane $\mathscr{B}$ from Theorem 1.1, establishing the geometric construction as valid and unique.