i'm trying to solve these problems:
(1) Suppose that $X\subset\mathbb{P}^2_\mathbb{C}$ is a smooth conic and $A,B\in X$ are two points on it. Find automorphisms of $\mathbb{P}^2_\mathbb{C}$ fixing $A,B$ and $X$ having infinite order.
(2) Suppose that $X\subset\mathbb{P}^3_\mathbb{C}$ is a smooth quadric surface. Let $D=X\cap Y$ a divisor which is a section with another smooth quadric surface. Find automorphisms of $\mathbb{P}^3_\mathbb{C}$ fixing $D$ and $X$ having infinite order.
For the first problem: i consider $C$ the point of intersection of the two tangents in $A$ and $B$ to $X$. The conic belong to the pencil of conics generated by $$\det(A|C|x)\det(B|C|x)=0\text{ and }\det(A|B|x)^2=0$$ If we consider $A,B,C$ as vectors in $\mathbb{C}^3$ we can choose automorphisms for every triple $(a,b,c)\in \mathbb{C}^{*3}$ such that c^2=ab, by supposing that $$T(A)=aA,T(B)=bB,T(C)=cC$$ So if $$X:\det(A|C|x)\det(B|C|x)=\alpha\det(A|B|x)^2$$ one can easily verify that $x\in X\Rightarrow T(x)\in X$
For the second problem i have tried to see the quadric as a member of a particular pencil in a similar way to problem 1, but i've not found a solution. Have you any idea?