roups nor virtually affine groups.

This paper is organized as follows: in Section \[s:recall\] we review some general facts and introduce the notation used throughout the text. In Section \[s:gever\] we describe some properties of the Veronese curve which are useful for our purposes. In Section \[s:chvh\] we characterize the complex hyperbolic subgroups that leave invariant a Veronese curve. In Section \[s:riv\] we depict those real hyperbolic subgroups leaving invariant a Veronese curve. Finally, in Section \[s:rep\] we show that every discrete compact surface group in $\PO(2,1)^+$ admits a deformation in $\PSL(3,\Bbb{C})$ which is not conjugate to a complex hyperbolic subgroup and has non-empty Kulkarni region of discontinuity.

Preliminaries {#s:recall}
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Projective geometry
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The complex projective space $\mathbb{P}^2_{\mathbb {C}}$ is defined as $$\mathbb{P}^{2}_{\mathbb {C}}=(\mathbb {C}^{3}\setminus \{0\})/\Bbb{C}^*,$$ where $\Bbb{C}^*$ acts by the usual scalar multiplication. This is a compact connected complex $2$-dimensional manifold. If $[\mbox{}]:\mathbb{C}^{3}\setminus\{0\}\rightarrow\mathbb{P}^{2}_{\mathbb{C}}$ is the quotient map, then a non-empty set $H\subset\mathbb{P}^2_{\mathbb{C}}$ is said to be a line if there is a $\mathbb{C}$-linear subspace $\widetilde{H}$ in $\mathbb{C}^{3}$ of dimension $2$ such that $[\widetilde{H}\setminus \{0\}]=H$. If $p,q$ are distinct points then $\overleftrightarrow{p,q}$ is the unique complex line passing through them. In this article, $e_1,e_2,e_{3}$ will denote the standard basis for $\Bbb{C}^{3}$.

 Projective transformations 
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The group of projective automorphisms of $\mathbb{P}^{2}_{\mathbb{C}}$ is defined as $$\PSL(3, \mathbb {C}) \,:=\, \GL({3}, \Bbb{C})/\Bbb{C}^*,$$ where $\Bbb{C}^*$ acts by the usual scalar multiplication. Then $\PSL(3, \mathbb{C})$ is a Lie group acting by biholomorphisms on $\Bbb{P}^2_{\Bbb{C}}$; its elements are called projective transformations. We denote by $[[\mbox{  }]]