Let S be a complex projective nonsingular algebraic surface of geometric genus pg(S) and irregularity q(S). The aim of this paper is to prove the following (1.1) THEOREM. Let A c S be a smooth curve which is an ample divisor and has genus g(A) = q(S). Assume furthermore that A 2 ~ q(S) -1, if pg(S) >~ 2; then (S, A) is one of the following pairs:
(i) (p2, line), (p2, conic), (Pa-bundle, section), (ii) (J(C), F) where J(C) is the Jacobian of a smooth curve C of genus two and F is C embedded in its Jacobian, up to a translation, or
As is known, if H c S is a smooth curve which is a very ample divisor, then the equality g(H) = q(S) holds if and only if (S, H) is one of the pairs listed in (i) (e.g. cf. [9, p. 388]). This is just one of the characterizations of the pairs (S, H) in (i) that have been given from many different points of view: the 2-connectedness of the divisors in the linear system [H[ [11], the vanishing cycles of a Lefschetz pencil containing H [5], the projective characters of S linearly normally embedded in a projective space via ]Hi ([3], [5]).
For pairs (S, A), A being a smooth curve contained in S as an ample divisor, in [6, Th. 2.2] it is shown that, under the additional assumption that h°((gs(A)) >~ 2, whenever q(S) >~ 2, the pairs (S, A) with g(A) = q(S) are exactly those listed in (i). The example (J(C), C) of a smooth curve of genus 2 embedded in its Jacobian was communicated to M. Patleschi and me by Prof. A. J. Sommese. Of course h°((gs(c)(C)) = 1. It seems reasonable to conjecture that the above pair (J(C), C) is essentially the unique to add to the list (i) when one considers the general situation for pairs (S, A) with no additional assumptions. Theorem (1.1) says only that this is true if AZ>>. q(S)-1 whenever pg(S) >>. 2; this assumption is crucial in the proof, but unfortunately no counterexample was found which could prove its necessity for the conjecture to be true.
Notice that this conjecture is far from being true in the wider context of * The author is a member of the G.N.S.A.G.A of the Italian C.N.R.