we will also show that tor quartic surfaces a disjoint line always exists and give a procedure to find it.
Our algorithm, starting from a polynomial equation with rational coefficients of a surface, returns as output a list of as many integers as the connected components of S, where the integers represent the ranks of the groups Hi relative to the various connected components. Our strategy will consist in reconstructing in a finite number of steps the homology of S using as a basic tool Morse theory (see for instance Milnor (1963) or Hirsch (1976)), as first proposed in Gianni and Traverso (1983). The basic step, to be used iteratively, is described in Section 4, while the main algorithm is presented in Section 5.
We have developed algorithms for performing all the tasks and we have produced a working implementation in Axiom of the entire procedure. To do this we assume that the given system of coordinates is a "good frame", in the sense defined in Section 2, which is always true up to a linear change of coordinates. In Section 6 we will describe algorithms to test whether the hypotheses both on the surface and on the system of coordinates are fulfilled or not. In this paper we do not deal with the complexity aspects of our algorithms.
For a proof of the classical results about surfaces, homotopy and homology already recalled and of those we will use throughout the paper, we refer to Massey (1977) or to Massey (1991).
2. First definitions and notations
Let ф(х, у, z, t) be a square-free homogeneous polynomial of degree d with rational coefficients defining a non-singular orientable real algebraic projective surface; since the surface is orientable, d has to be even. By an abuse of language, by the same term "surface" we will indicate also the set S of the points in RP3 satisfying the equation <j>(x,y,z,t) =0.
Let us emphasize that, when we say that the surface S is non-singular, we mean that no point in RP3 annihilates ф and all its first partial derivatives. Note that, since S can be seen as the real part of the complex zero-set Sc of ф in CP3, according to our assumptions Sc may contain non-real singular points.
Recall that, if / : S ->■ К is a differentiable function defined on a smooth surface, a point P e S is said to be critical for / if, working in local coordinates u\,U2, both first partial derivatives of / vanish at P; then f(P) is said to be a critical value. P is said to be non-degenerate if the Hessian form of / at P is a non-degenerate quadratic form, i.e. the 2x2 matrix Hp = (d2f/d uiduj)(P) is invertible; the number of negative eigenvalues of Hp is called the index of the critical point P. If all critical points of / are non-degenerate, / is called a Morse function.
In this paper we will describe an algorithm to study the topology of S = {ф(х, у, z, t) = 0} assuming that there exists a line L с KP3 such that LC\ S = 0 and that in our system of homogeneous coordinates [x, y, z, t] we have
(1) L = {z = oJ = o}
(2) If Z = {x = 0, у = 0}, the function ж : S -* Z = ЮР1, restriction to S of the projection p : KP3\L —► Z of ЮР3 onto Z with centre L, is a Morse function and [0, 0, 1,0] is not a critical value for ж
(3) Whenever P, Q e S are critical points for ж, we have ж(Р) ф n(Q).
