On the Bicanonical Sets of a Certain Class of Curves

Extract

W. G. Welchman in his work on fundamental scrolls* obtains as directrix curves to such scrolls a canonical curve ^pK^2p−2 in [p − 1] and a non-special curve ^pCⁿ in [n − p]. These latter curves may not, however, be general curves regarded projectively, and it is an interesting question to find out the geometrical interpretation of their particularity. The curves C and K are, of course, in birational correspondence, and the prime sections of C correspond to the sections of K by quadrics through a contact set*, i.e. a set of points such that there is a quadric which touches K at every point of the set. For k small enough it is clear that every set of k points is a contact set and in this case the curves C are quite general. For larger k the fact that the set is a contact set simply means that, on C, the points of the bicanonical sets residual to a prime section lie themselves in primes (when k is sufficiently small the number of points in this residual set is such that they always lie in a prime).