# Local embedding dimension of a curve

In the paper Random hypersurfaces and embedding curves in finite fields by Joseph Gunther, I am trying to understand the following example

Let $$C$$ be the rational curve defined in $$\mathbb{P}^3_{\mathbb{F}_q}$$ by $$w = 0$$ and $$y^2z−x^3 +x^2z = 0$$. Then $$ζ_{V_1}(s)^{-1}= \frac{1-q^{1-s}}{1-q^{-s}}, \ ζ_{V_2}(s)^{−1} =1−q^{−s}$$ and $$ζ_{X-V}(s)^{−1} =(1−q^{−s})(1−q^{2−s})(1−q^{3−s}).$$

Here $$V_i \ (i=1,2)$$, is the (locally closed) subset of $$C$$ whose closed points are exactly those of local embedding dimension $$i$$ in $$C$$. And $$ζ$$ is the zeta function of a variety.

I can't figure out what exactly $$V_1$$ and $$V_2$$ are, so I can calculate their zeta function. Is there an explicit way to calculate $$V_i$$ for a general projective scheme given by homogeneous polynomials?

The local embedding dimension of a closed point $$x$$ is just the dimension of the tangent space at $$x$$ over the residue field $$\kappa(x)$$, so in the case of this curve, $$V_1$$ is the set of regular points and $$V_2$$ is the set of singular points with 2-dimensional tangent space, aka all the singular points for this curve because it's planar. (See section 3 for the definition of "local embedding dimension" in this paper - this matches most other places I've seen this in the literature.)
• I need to calculate the degree of those singular and regular points in order to compute the zeta function. How do I do that? Looking at the zeta function of $V_2$ it seems that $C$ has only one singular point of degree $1$. Is this true? We are working with finite fields, so I don't know how to proceed. – I like Cake Apr 15 '20 at 22:00
• I was making my life difficult. There is only one singular point and the rest are regular points, so the zeta function of $V_1$ is just zeta function of $C$ divided by the zeta function of this singular point. And the way you calculate the zeta function of $C$ is by calculating $|C(\mathbb{F}_{q^n})|$. Is this the correct approach? – I like Cake Apr 15 '20 at 23:13