Let π be a projective plane of order q and let K be a subset of its point-set, |K| = k; the set K will be called a two character k-set of type (m,n) (or briefly a k-set of type (m,n)) if a line in π either meets K in m or in n points and m-secants and n-secants actually exist; hence 0 ≤ m < n ≤ q + l.
In 1966, Tallini Scafati [6] gave necessary arithmetical conditions for a k-set of type (m,n) to exist in a projective plane of order q, q = ph, p a prime, h a non-negative integer. Namely, she proved that k must be a root of the equation
k2 - k(n + m + q(n + m - 1)) + mn(q2 + q+1) = 0, (1)
so that its discriminant must be a non-negative square; moreover, n-m must divide q. As a special case, she characterized k-sets of type (l,n), n ≠ q + 1, proving that q must be a square, n = √q +1 and such an arc either is a Baer subplane or a Hermitian arc (a Hermitian curve if a certain reciprocity condition holds). Thus sets of type (m,q), m ≠ 0, are also characterized, being the complements of the preceding ones. A set of type (1,q+1) is a line; a set of type (0,q) is an affine plane.