Ruled surfaces have been studied by NAGATA IS], MARUYAMA [3, 41 and other authors from the point of view of classification. Especially on rational ruled surfaces we have known many facts, for example, an explicit condition for a divisor D to be ample, that for ID( to have an irreducible member and so on. While we can calculate the dimension of (DI and the genus of a general member of (Dl for an ample divisor D, we know little about special members of ID[. In this paper we compute the dimension of the family of all irreducible reduced rational curves of ID[, examining morphisms from P' to a rational ruled surface F, (Theorem 5.1).
We are also interested in morphisms from F,. to F, and those from F, to P2. One can ask what sorts of surjective morphisms exist from F,. to F,. Although automorphism groups of F, are studied by MARUYAMA [3], the author knows no previous work from this point of view. This paper gives a definite answer to the question (Proposition 2.1).
Let 71 : X -+ P' be a rational ruled surface with invariant e defined over an algebraically closed field k.
The characteristic of k is arbitrary unless specifically mentioned. Following the notation of HARTSHORNE [llf is a fibre, C , is a section with self-intersection numbere and Pic X = ZC, @ Zf. We remark that if e > 0, then C , is unique. If X ' is a rational ruled surface, then TC', e', C & and f ' mean the same as above. Let y : P -5 P' be a finite morphism. We say a morphism cp : X ' + X is over y , when cp satisfies TCO cp = y 071'. We understand that the image of a morphism has always the reduced structure.
In the first section some fundamental facts on rational ruled surfaces are stated.
In 0 2 we investigate a morphism cp from F,. to F, such that the image is not a fibre.
Proposition 2.1 states that if cp is surjective, then cp is finite and there is a finite morphism w such that cp is over y . Moreover there are integers (a, d) with ae' = de such that q*C0 = aCh and cp*f= df'. Here a is also equal to the degree of the morphism cplf. from f ' to its image, and d is equal to the degree of the morphism cplc, from C , to its image. We denote them by deg(f'/cp(f')) and deg (CJcp(Cb)) respectively. Conversely for any (a, d) with ae' = de, and for any finite morphism y on P' of degree d, there are finite morphisms over y . These results are summerized in Theorems 2.2 and 2.3. Let t , 2 1 and t , 2 1 be the least numbers such that anl -tl(x,y) # 0, b,,, -t 2 ( ~, y ) # 0 respectively. We may assume t , I t,, then t = E, and gn-t(X,Y)=an,-t.bn, + a n l -t + l .bn,-l + ... +a,, .bn2-r a,, -t . bn2, if t < t, anl -*. bn2 + a,, . b,,-,, if t = t,. In both cases gn-t(x, y ) is divided by ax + by, which is a contradiction.