I know we can bound the triple point on quintics in cp^3 by 5. But how to write down quintics with 5 ordinary triple point (here are simple elliptic singularity)explicitly? 



example of quintics with 5 ordinary triple point
Choose five points $P\_i$ in general linear position (all choices are
equivalent under ${\rm PGL}\_4$, so you might as well put four of them
at the coordinate vectors and the fifth at $(1:1:1:1)$); then
at each $P\_i$ the condition of a triple point imposes $1+3+6 = 10$
linear conditions on the space of quintics, which has dimension
${8 \choose 5} = 56$, so you get a linear system of quintics 
with a triple point at each $P\_i$, of projective dimension
$56 - 5\cdot 10 - 1 = 5$.


A quick Google search turns up a paper



> 
> Stephan Endrass, Ulf Persson, and Jan Stevens: 
>  Surfaces with triple points,
>  *J. Algebraic Geom.* **12** (2003), 307--320.
>  arXiv: [math/0010163](https://arxiv.org/abs/math/0010163)
> 
> 
> 


which gives on pages 5-6 an alternative description. Start with a
cubic surface that has a triple point (so is a cone over a cubic 
plane curve), and apply a "reciprocal transformation": choose four points
$p\_i$ on the cubic (other than the cubic singularity), use projective coordinates $(x\_1:x\_2:x\_3:x\_4)$ that make $p\_i$ 
the coordinate points, and apply the birational involution
$$
(x\_1:x\_2:x\_3:x\_4) \leftarrow - \ - \rightarrow (1/x\_1:1/x\_2:1/x\_3:1/x\_4).
$$
As remarked at the end of this section, "The above analysis is very elementary, 
and parts of it have not too surprisingly already appeared in the literature",
citing a 1952 paper in Italian:



> 
> D. Gallarati, Sulle superficie del quinto ordine dotate di punti tripli,
>  *Rend. Accad. Naz. Lincei, serie VIII*, **XII** (1952), 70--75.
> 
> 
> 


