Consider $C$ in $\mathbf{P}^3_k = \mathrm{Proj}[x_0,...,x_3]$ defined by $$x_0x_3 - x_1^2 = 0$$ and $$x_0^2 + x_2^2 - x_3^2 = 0$$ where $k$ is an algebraically closed field. Why is this curve nonsingular?
I've been trying work this out in scheme-theoretic terms; in particular, is it sufficient to show that if $H$ is the homogeneous coordinate ring of $C,$ that on the rings $(H_f)_0$ (i.e. the zero-degree subrings of $H$ localized at $f$) corresponding to basic affine opens of $C$ that the stalks $((H_f)_0)_\mathfrak{p}$ are discrete valuation rings i.e. UFDs with unique-up-to-unit irreducibles?
(And if so, what's the best way of doing this?)
Any help, clarification, or correction would be greatly appreciated.
Yours may be a case of excess of technology...
You probably know that to check non-singularity it is enough to do it locally, so you can consider, for example, the standard open covering of $P^3$. Then you are in affine $3$-space, and you can use the Jacobian criterion.
For example, in the set where $x_0\neq0$ we can take affine coordinates $x=x_1/x_0$, $y=x_2/x_0$ ad $z=x_3/x_0$, and the equations of the intersection of your curve are $$z-x^2=0, \qquad 1+y^2-z^2=0.$$ Can you prove this is non-singular?