In the paper ["Normal Subgroups in the Cremona Group"](https://link.springer.com/content/pdf/10.1007/s11511-013-0090-1.pdf), under remark 5.1 they stated that for any generic set $\Sigma \subset \mathbb{P}^2\_\mathbb{C}$ of $k$ points, and $h$ is an automorphism of $\mathbb{P}^2\_\mathbb{C}$, then $h$ is the identity as soon $h(\Sigma)\cap \Sigma$ contains at least 5 points.


Can anyone be kind enough to show how do I prove it or are there any papers proving this result? 


Thank you very much



$h$ is identity as soon as $h(\Sigma)\cap \Sigma$ contains at least 5 points
In other words, if $\Sigma$ is generic, there does not exist a non-identity automorphism $h$ for which $|\Sigma\cap h(\Sigma)|\geqslant 5$. Denote $\Sigma=\{p\_1,\dots,p\_k\}$ and assume that $h(p\_i)=p\_{f(i)}$ for $i=1,2,3,4,5$ and a certain (we may supposed that fixed) injective function $f:\{1,2,3,4,5\}\rightarrow \{1,2,\dots,k\}$. Consider two cases.


1) Assume that some $f(i)$ is greater than 5, say, $f(5)\notin \{1,2,3,4,5\}$. Then if we fix all points except $p\_{f(5)}$, $h$ is uniquely determined by $h(p\_i)=p\_{f(i)}$ for $i=1,2,3,4$, and the event $h(p\_5)=p\_{f(5)}$ generically does not hold. 


2) $f$ is a bijection of the set $\{1,2,3,4,5\}$, but not identity permutation (if $f$ is identity, then $h$ is identity). 


2.1) If $f$ has a fixed point, say $f(5)=5$, again $h$ is determined by $h(p\_i)=p\_{f(i)}$ for $i=1,2,3,4$ (and $h$ is not identity), and the event that $p\_5$ is a fixed point of $h$ generically does not hold.


2.2) $f$ is a cycle of length 5 or a product of disjoint cycles of lengths 3 and 2. The 5-tuple $\{p\_1,p\_2,p\_3,p\_4,p\_5\}$ is projectively equivalent to the 5-tuple $\{p\_{f(1)},p\_{f(2)},p\_{f(3)},p\_{f(4)}, p\_{f(5)}\}$. Take your favourite projective invariant of a 5-tuple, like the cross ratio of the points $p\_1,p\_2,p\_3p\_4\cap p\_1p\_2,p\_3p\_5\cap p\_1p\_2$ and verify that generically it changes under the permutation $f$ (it suffices to find one specific 5-tuple of points for which it changes).


