Arc (projective geometry)
$k$-arcs in projective planes
A -arc in the Desarguesian projective plane
over the Galois field of order
is a set of
points, no three of which are collinear. It is immediate that
, but if
is odd, then
. The classical example of a
-arc is a conic, that is, a set of points projectively equivalent to
. If
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
-arc. A
-arc in
is called an oval and a
-arc in
,
even, is called a hyperoval (cf. also Oval).
In his celebrated 1955 theorem, B. Segre proved that in ,
odd, every
-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 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
; the case
still relies on a computer search.
$k$-arcs in projective space
A -arc in the
-dimensional projective space
is a set of
points with
and at most
in each hyperplane. (This definition coincides with the definition of
-arc in
given above.) Further, a
-arc of
,
and
, exists if and only if a
-arc of
exists. A linear maximum distance separable code is a linear code of length
, dimension
and minimum distance
(cf. also Coding and decoding). It is well-known that for
these notions are equivalent; as each can be viewed as a set of
vectors in an
-dimensional vector space over
with each
vectors being linearly independent.
The classical example of a -arc in
is a normal rational curve, that is, a set of points projectively equivalent to
. The only known (1998) non-classical examples are examples in
, for
even, constructed by L.R.A. Casse and D.G. Glynn, and a
-arc in
constructed by Glynn. The main open (1998) problem in the area is the resolution of the so-called main conjecture for
-arcs and maximum distance separable codes, which is that if
, then the size of a largest
-arc in
is
if
is even and
or
, and is
in all other cases. The main conjecture has been settled for
and in a number of further cases. It is also of great interest to characterize the largest-size
-arcs, and to determine the size of the second-largest complete
-arcs in
, where a
-arc of
is complete if it is contained in no
-arc of
.
$(k,n)$-arcs in $\mathrm{PG}(2,q)$
For , a
-arc (or arc of degree
) in
is a set
of
points such that each line meets
in at most
points and there is a line meeting
in exactly
points. It is immediate that
with equality if and only if each line meets
in 0 or
points. Equality also implies that either
or
divides
. A
-arc with
is called a maximal arc of degree
, and is non-trivial if
. If
is even, there are examples of non-trivial maximal arcs of degree
for every
dividing
(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
where
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 -arc,
-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 |
Arc (projective geometry). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arc_(projective_geometry)&oldid=25358