Finite flag-transitive semiovals

A linear space is an incidence structure of points and lines such that any two points are incident with exactly one line, any point being incident with at least two lines and any line with at least two points. Within a linear space S, we shall often identify a line L with the set of points incident with L and define the size of L to be the number of points of S incident with L. The degree of a point x of S is the number of lines of S through x. S is called regular, or more precisely (u, k)-regular, if it has a finite number u of points and if all its lines have the same size k (such spaces are 2 (u, k, 1) designs). Let P be a projective plane with point-set P. Given a subset S of P, a line L of P is called exterior, tangent or secant according as 1 L n SI = 0, 1 or 22. If S is not contained in a line of P, we denote by S the linear space induced on S by P: its points are those of S and its lines are the (intersections with S of the) secants. S is a semioval [4] iff S is nonempty and every point of S is on exactly one tangent. An oval (resp. thick semioual) is a semioval meeting all secants in only two (resp. at least three) points. Note that if S is a thick semioval, then for every x E S the set S\ (x> is still a semioval. A semioval S is (u, k)-regular iff S is (u, k)-regular. The set S of absolute points of a Hermitian polarity in PG(2, q2) is a thick (q3 + 1, q + 1 )-regular semioval, the linear space S is then called a Hermitian unital of order q and will be denoted by U,(q). A flag of a semioval S is an incident point-line pair of S. We investigate here the finite flag-transitive semiovals, or more precisely the triples (P, S, G), where S is a semioval in a projective plane P and G < Aut P