Let us take the second degree Hirzebruch surface F_2 which is a holomorphic CP^1 bundle over CP^1 having sections of self intersections +2 and -2. Let me denote the class of the -2 section by C and class of the sphere fiber by F. Picard group of F_2 is generated by C and F.
1) It is known that the linear system |2C+5F| is very ample and it contains an irreducible curve, let me denote it by 2C+5F. On the other hand, I take the linear system |C+2F|. By the same proposition in Hartshorne, this system is not very ample but contains an irreducible curve, denoted by C+2F.
The curves (2C+5F) and (C+2F) intersect at 5 points generically. My question is : Can we move 2C+5F so that it intersects C+2F at one point with multiplicity 5? But while moving I do not want to change the genera and self intersections. Or let me put it another way; is there a member of the very ample system |2C+5F| which intersects a member of |C+2F| at one point with multiplicity 5?
As I know in very ample systems there is more freedom, since they are not rigid. So I was hoping the curve 2C+5F in the way that I want, but I could not find any proof in relation to these.
2) My second question: The fiber F and the -2 section C of F_2 intersect once, say, at the point p. Can we find a member of the very ample linear system |2C+5F| which passes through this point p such that, at p, C and 2C+5F intersect once, and F and 2C+5F intersect once but with multiplicity 2?
I would appreciate any suggestions or resources. Thanks.