Local embedding dimension of a curve

Question

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?

Answer 1

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.)

In general, the Jacobian matrix tells you how to calculate this dimension. This is widely covered and "well known" enough that you should google it and/or look it up in your favorite book.