Let us take the second degree Hirzebruch surface $\mathbb{F}_2$ which is a holomorphic $\mathbb{CP}^1$ bundle over $\mathbb{CP}^1$ 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 $F$. The Picard group of $\mathbb{F}_2$ is generated by $C$ and $F$.
- 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.
- My second question: The fiber $F$ and the $-2$ section $C$ of $\mathbb{F}_2$ 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.