The classical concept of a conic leads in a natural way to the concept of an oval in an arbitrary projective plane: An oval is a subset Ω of points satisfying both of the following properties: i) no three points of Ω are col linear; ii) Q has exactly one 1-secant (also called a tangent) at each one of its points. If the plane is finite and has order n, then an oval consists of n+1 points.

Ovals of finite projective planes have been intensively studied since 1954. The starting point was the famous theorem of B. Segre [94], [95]: In a Desarguesian plane of odd order, the ovals are exactly the irreducible conies.

This paper is a survey of known results in the following areas:

1) The classification problem for ovals in a desarguesian plane of even order.

2) Ovals in finite non desarguesian planes.

3) Pascal's theorem for ovals and abstract ovals.

4) Collineation groups fixing an oval; some characterizations of the finite desarguesian planes.

THE CLASSIFICATION PROBLEM FOR OVALS IN A DESARGUESIAN PLANE OF EVEN ORDER

In 1956 Segre pointed out that his result on the characterization of conies cannot be extended to desarguesian planes of even order. The classification of ovals in these planes is still an open problem and seems to be very complex.

We give a brief account of the known ovals in desarguesian planes of even order, but for detailed information concerning the extensive theory of ovals developed by Segre and his school the reader is referred in particular to the books [53], [98]. Quite recently, some new investigations have been carried out. Details will be found in the survey papers [15], [80].

It was on March 20, 1984, that I wrote to Herb Ryser and proposed that we write together a book on the subject of combinatorial matrix theory. He wrote back nine days later that “I am greatly intrigued by the idea of writing a joint book with you on combinatorial matrix theory. … Ideally, such a book would contain lots of information but not be cluttered with detail. Above all it should reveal the great power and beauty of matrix theory in combinatorial settings. … I do believe that we could come up with a really exciting and elegant book that could have a great deal of impact. Let me say once again that at this time I am greatly intrigued by the whole idea.” We met that summer at the small Combinatorial Matrix Theory Workshop held in Opinicon (Ontario, Canada) and had some discussions about what might go into the book, its style, a timetable for completing it, and so forth. In the next year we discussed our ideas somewhat more and exchanged some preliminary material for the book. We also made plans for me to come out to Caltech in January, 1986, for six months in order that we could really work on the book. Those were exciting days filled with enthusiasm and great anticipation.

Herb Ryser died on July 12, 1985. His death was a big loss for me. Strange as it may sound, I was angry. Angry because Herb was greatly looking forward to his imminent retirement from Caltech and to our working together on the book. In spite of his death and as previously arranged, I went to Caltech in January of 1986 and did some work on the book, writing preliminary versions of what are now Chapters 1, 2, 3, 4, 5 and 6. As I have been writing these last couple of years, it has become clear that the book we had envisioned, a book of about 300 pages covering the basic results and methods of combinatorial matrix theory, was not realistic.