Arc (projective geometry)

$k$-arcs in projective planes

A $k$-arc in the Desarguesian projective plane $\operatorname{PG} ( 2 , q )$ over the Galois field of order $q$ is a set of $k$ points, no three of which are collinear. It is immediate that $k \leq q + 2$, but if $q$ is odd, then $k \leq q + 1$. The classical example of a $( q + 1 )$-arc is a conic, that is, a set of points projectively equivalent to $\{ ( 1 , t , t ^ { 2 } ) : t \in \operatorname {GF} ( q ) \} \cup \{ ( 0,0,1 ) \}$. If $q$ is even, then all the tangents to the conic pass through a common point, called the nucleus; hence the set of points of a conic together with the nucleus is a $( q + 2 )$-arc. A $( q + 1 )$-arc in $\operatorname{PG} ( 2 , q )$ is called an oval and a $( q + 2 )$-arc in $\operatorname{PG} ( 2 , q )$, $q$ even, is called a hyperoval (cf. also Oval).

In his celebrated 1955 theorem, B. Segre proved that in $\operatorname{PG} ( 2 , q )$, $q$ odd, every $( q + 1 )$-arc is a conic. This important result, linking combinatorial and algebraic properties of sets of points, was of great importance in the early development of the field of finite geometry, and many results in the same spirit have been proved.

The situation when $q$ is even is quite different. Apart from the "classical" examples provided by a conic together with its nucleus mentioned above, there are currently (1998) seven infinite families of hyperovals known, and several examples which do not at present fit into any known infinite family. The classification of hyperovals is known only for $q \leq 32$; the case $q = 32$ still relies on a computer search.

$k$-arcs in projective space

A $k$-arc in the $n$-dimensional projective space $\operatorname{PG} ( n , q )$ is a set of $k$ points with $k \geq n + 1$ and at most $n$ in each hyperplane. (This definition coincides with the definition of $k$-arc in $\operatorname{PG} ( 2 , q )$ given above.) Further, a $k$-arc of $\operatorname{PG} ( n , q )$, $n \geq 2$ and $k \geq n + 4$, exists if and only if a $k$-arc of $\operatorname{PG} ( k - n - 2 , q )$ exists. A linear maximum distance separable code is a linear code of length $k$, dimension $n + 1$ and minimum distance $d = k - n + 2$ (cf. also Coding and decoding). It is well-known that for $k \geq n + 1$ these notions are equivalent; as each can be viewed as a set of $k$ vectors in an $( n + 1 )$-dimensional vector space over $\operatorname{GF} ( q ),$ with each $n + 1$ vectors being linearly independent.

The classical example of a $( q + 1 )$-arc in $\operatorname{PG} ( n , q )$ is a normal rational curve, that is, a set of points projectively equivalent to $\{ ( 1 , t , t ^ { 2 } , \dots , t ^ { n } ) : t \in \operatorname{GF} ( q ) \} \cup \{ ( 0 , \dots , 0,1 ) \}$. The only known (1998) non-classical examples are examples in $\operatorname{PG} ( 3 , q )$, for $q$ even, constructed by L.R.A. Casse and D.G. Glynn, and a $10$-arc in $\operatorname{PG} ( 4,9 )$ constructed by Glynn. The main open (1998) problem in the area is the resolution of the so-called main conjecture for $k$-arcs and maximum distance separable codes, which is that if $q > n + 1$, then the size of a largest $k$-arc in $\operatorname{PG} ( n , q )$ is $q + 2$ if $q$ is even and $n = 2$ or $n = q - 2$, and is $q + 1$ in all other cases. The main conjecture has been settled for $n=2,3,4$ and in a number of further cases. It is also of great interest to characterize the largest-size $k$-arcs, and to determine the size of the second-largest complete $k$-arcs in $\operatorname{PG} ( n , q )$, where a $k$-arc of $\operatorname{PG} ( n , q )$ is complete if it is contained in no $( k + 1 )$-arc of $\operatorname{PG} ( n , q )$.

$(k,n)$-arcs in $\mathrm{PG}(2,q)$

For $n \geq 2$, a $( k , n )$-arc (or arc of degree $n$) in $\operatorname{PG} ( 2 , q )$ is a set $\mathcal{K}$ of $k$ points such that each line meets $\mathcal{K}$ in at most $n$ points and there is a line meeting $\mathcal{K}$ in exactly $n$ points. It is immediate that $k \leq ( n - 1 ) q + n,$ with equality if and only if each line meets $\mathcal{K}$ in 0 or $n$ points. Equality also implies that either $n = q + 1$ or $n$ divides $q$. A $( k , n )$-arc with $k = ( n - 1 ) q + n$ is called a maximal arc of degree $n$, and is non-trivial if $2 \leq n \leq q - 1$. If $q$ is even, there are examples of non-trivial maximal arcs of degree $n$ for every $n$ dividing $q$ (due to R.H.F. Denniston and J.A. Thas). On the other hand, S. Ball, A. Blokhuis and F. Mazzocca have shown that non-trivial maximal arcs in $\operatorname {PG} ( 2 , q ),$ where $q$ is odd, do not exist. (The proof appears in [a2].)

See [a3] for a survey on each topic mentioned above; for a comprehensive account including more details, results and the references, see [a2], Chaps. 8, 9.

The definitions of $k$-arc, $( k , n )$-arc, oval, hyperoval and maximal arc in a non–Desarguesian projective plane are the combinatorial definitions given above, but in this case there are relatively few examples and the theory is not so well-developed.

References

[a1]	J.W.P. Hirschfeld, J.A. Thas, "General Galois geometries" , Oxford Univ. Press (1991)
[a2]	J.W.P. Hirschfeld, "Projective geometries over finite fields" , Oxford Univ. Press (1998) (Edition: Second)
[a3]	J.A. Thas, "Projective geometry over a finite field" F. Buekenhout (ed.) , Handbook of Incidence Geometry, Buildings and Foundations , Elsevier (1995) pp. Chap. 7; 295–348