Equality and Maximal Arcs

4.1 The dictionary

Let $\mathcal L$ be an $r$-fold configuration and $S=\delta(\mathcal L)$ its dual point set (§2.2): $O\notin S$, $|S|=r(q+1)$, and every line through $O$ contains exactly $r$ points of $S$. For a line $L$ not through $O$, $|L\cap S|$ is the multiplicity of the affine point $\delta(L)$, and (2.1) says $|U(\mathcal L)|$ is the number of lines not through $O$ that meet $S$.

Lemma 4.1. Let $M=S\cup\{O\}$. Then every line through $O$ meets $M$ in exactly $r+1$ points, and for a line $L$ not through $O$, $|L\cap M|=|L\cap S|=m_{\delta(L)}$. In particular: every point of $U(\mathcal L)$ has multiplicity exactly $r+1$ if and only if $M$ is a maximal arc of degree $r+1$.

Proof. The first two statements are immediate from the definition of $S$. If every $m_P\in\{0,r+1\}$ (multiplicity $0$ meaning $P\notin U$), then every line meets $M$ in $0$ or $r+1$ points, and $M\ne\emptyset$; conversely if $M$ is a maximal arc of degree $r+1$ then each $|L\cap S|\in\{0,r+1\}$ for $L\not\ni O$, i.e. each $m_P\in\{0,r+1\}$. ∎

The size check is reassuring: a maximal arc of degree $n=r+1$ has $(n-1)q+n=rq+r+1=r(q+1)+1=|S|+1$ points (Lemma 2.3).

4.2 The equality theorem

Theorem 4.2 (Theorem B). An $r$-fold configuration $\mathcal L$ attains the bound $|U(\mathcal L)|=\frac r{r+1}q(q+1)$ if and only if $\delta(\mathcal L)\cup\{O\}$ is a maximal arc of degree $r+1$ in $PG(2,q)$. Conversely, if $M$ is a maximal arc of degree $r+1$ and $O$ is any point of $M$, then $\delta(M\setminus\{O\})$, computed in coordinates in which $O=(0:0:1)$, is an $r$-fold configuration attaining the bound.

Proof. The first statement is Theorem 3.2 combined with Lemma 4.1. For the converse, choose a projective coordinate system with $O=(0:0:1)$; then every line through $O$ meets $M$ in $r+1$ points, so $S=M\setminus\{O\}$ has exactly $r$ points on each line through $O$ and $O\notin S$, and $\delta(S)$ is an $r$-fold configuration. Its multiplicities are the values $|L\cap S|$, $L\not\ni O$, which are $0$ or $r+1$; so every point of its union has multiplicity $r+1$ and Theorem 3.2 gives equality. ∎

Corollary 4.3. $\kappa_r(q)=\frac r{r+1}q(q+1)$ if and only if $r\in\{q-1,q\}$, or $q$ is even and $(r+1)\mid q$. In all other cases, and in particular for every odd $q$ and every $1\le r\le q-2$, $\kappa_r(q)>\frac r{r+1}q(q+1)$.

Proof. By Theorem 4.2, equality holds for some configuration iff a maximal arc of degree $n=r+1$ exists in $PG(2,q)$, with $2\le n\le q+1$. For $n=q+1$ the whole plane is such an arc; for $n=q$, the affine plane; these are $r=q$ and $r=q-1$. For $2\le n\le q-1$, existence holds iff $q$ is even and $n\mid q$, by Lemma 2.3, Theorem 2.4, and Theorem 2.6. ∎

Thus the equality cases of the counting bound are completely known, and the list is short. For $q=2^h$ they are $r+1\in\{2,4,\dots,2^h\}$ together with $r=q$; for $q$ odd, only $r\in\{q-1,q\}$. The strict inequality for odd $q$ rests on the Ball–Blokhuis–Mazzocca theorem and I know no elementary proof of it, even for $r=2$; the case $r=1$ is the classical statement that a $(q+2)$-set in $PG(2,q)$, $q$ odd, has three collinear points, which does have elementary proofs.

4.3 Examples

Dual hyperovals ($r=1$, $q$ even). A hyperoval $M$ through $O$ dualises to $q+1$ lines, one per direction, no three concurrent: $|U|=\binom{q+1}2$. This is the Blokhuis–Mazzocca minimum for even $q$, recovered.

Pencil complements ($r=q-1$). The affine plane $AG(2,q)$ is a maximal arc of degree $q$; choosing $O$ to be an affine point and dualising, one obtains all lines not through a fixed point $\delta(O)$, i.e. $q-1$ lines per direction, with union the plane minus that point: $|U|=q^2-1=\frac{q-1}q q(q+1)$. The same configuration is described directly in §7.1.

Dual Denniston arcs. For $q=2^h$ and $n\mid q$, the arc $K$ of Theorem 2.4 contains the origin, and dualising with $O$ the origin gives an $(n-1)$-fold configuration with $|U|=\frac{n-1}nq(q+1)$ and all multiplicities $n$. For $q=8$, $n=4$: $27$ lines, three per direction, union of $54$ points, every one on exactly four lines. This configuration was constructed explicitly (Appendix A.2) and its multiplicities verified. In odd order no such regularity is possible (Corollary 4.3); for comparison, the extremal $q=7$, $r=2$ configuration has points of multiplicity $1$, $2$, $3$, and $4$ (§6.4).

Sub-configurations. By Remark 2.5, a Denniston arc of degree $n$ contains Denniston arcs of every degree $n'\mid n$ through the origin, so the dual $(n-1)$-fold configuration contains $(n'-1)$-fold sub-configurations attaining their own bounds. A $3$-fold configuration in $AG(2,8)$ of the dual Denniston kind thus contains a dual hyperoval of $36$ points inside its $54$. Choosing $r$ lines per direction from the dual arc with $r+1\nmid n$ gives upper bounds for $\kappa_r(q)$ that turn out to be sharp in the one case that could be checked (§7.2).

4.4 Two remarks on the duality

First, the duality is not an isomorphism of extremal problems in general: the dual of an $r$-fold configuration has the specific form "$r$ points on each line of the pencil at $O$", and the quantity minimised is the number of lines meeting it, which is not a standard quantity for point sets. It becomes standard exactly in the equality case, where "no line meets $S\cup\{O\}$ in a number of points other than $0$ or $r+1$" is the definition of a maximal arc. This is why the equality case is fully understood while the near-equality cases are not: the theory of maximal arcs is rigid, and its stability version (how close to a maximal arc can a set be when no maximal arc exists?) is not part of it.

Second, the correspondence gives an interpretation of $\kappa_r(q)$ in the dual plane: $q^2-\kappa_r(q)$ is the largest number of lines not through $O$ that can be skew to a set with $r$ points on each line through $O$. For $r=1$ and $q$ odd, Blokhuis and Mazzocca's theorem says this is $(q-1)^2/2$, attained when the chosen lines are $q$ tangents of a conic together with one further line. In the dual plane $S$ then consists of $q$ points of a conic through $O$ together with a point $A\ne O$ on the tangent at $O$, and the skew lines are the $q(q-1)/2$ external lines of the conic other than the $(q-1)/2$ of them through $A$, which is how $(q-1)^2/2$ arises. For $r\ge2$ the maximisers are unknown beyond the computed cases.