coefficients for getting a smooth equation I'm studying the equation
$$ x_0q_0 + x_1q_1 + x_2q_2 = 0$$
where $q_i$ is a homogeneous polynomial of degree two in the variables $x_0,\ldots,x_5$.
I would like to have some simple choices for the coefficients (in $\mathbb{C}$) of the $q_i$ so that (away from zero) the partial derivatives do not vanish simultaneously.
any suggestions? I've tried a few ones but I was trying to make things too simple and ended up with critical points.
 A: Yes, this is one of those slightly frustrating things in algebraic geometry: according to the theory, you expect that a "general" object should have the properties you want, but any example you write down might not be general enough. I don't know a general (!) method for getting around this problem, but I'll say what I can.
First of all, the equation has to involve all the variables --- if it didn't, it would be a cone.
Next, the common zero locus of all these equations (that is, for all possible choices of $q_i$) is the linear subspace $L=\{x_0=x_1=x_2=0\}$. So Bertini's theorem guarantees that a general choice of $q_i$ will give you a variety that is at least smooth away from $L$. 
Let's not worry about what exactly "general" means here, and instead focus on the question of whether we can choose the $q_i$ to make our variety smooth along $L$. Taking partial derivatives, that means that we need to include terms in the $q_i$ that don't involve any of $x_0$, $x_1$, and $x_2$.
The simplest thing you could start with, then, would be to take say $q_0=x_3^2$. Since we need to involve all the variables, it's natural to take $q_1=x_4^2$ and $q_2=x_5^2$. However, it's immediately clear this won't work, because if we swap the roles of $x_0,x_1,x_2$ and $x_3,x_4,x_5$ in the previous paragraph, we see our variety will be singular along the linear space $x_3=x_4=x_5=0$. So there needs to be some mixing of the two sets of variables.
The next simplest thing would be to take $q_0=x_0^2+x_3^2$, and likewise $q_1=x_1^2+x_4^2$, $q_2=x_2^2+x_5^2$. Calculating partial derivatives, we get the following:
$$ (3x_0^2+x_3^2, \, 3x_1^2+x_4^2, \, 3x_2^2+x_5^2, \, 2x_0x_3, \, 2x_1x_4, \, 2x_2x_5) $$
and it's clear these can't all vanish at a point of $\mathbf{P}^5$. So this choice of the $q_i$ does what you want.
This was pretty long-winded, but I thought it might help more to explain one way to get to an answer, rather than just writing one down. 
