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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?

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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.

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  • $\begingroup$ 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. $\endgroup$ – I like Cake Apr 15 '20 at 22:00
  • $\begingroup$ Yes, this is true - an irreducible plane cubic can only have one singular point by Bezout. As per Abhyankar, there are only psychological difficulties in doing this sort of thing for finite fields. $\endgroup$ – KReiser Apr 15 '20 at 22:33
  • $\begingroup$ 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? $\endgroup$ – I like Cake Apr 15 '20 at 23:13
  • $\begingroup$ Yes, that's correct. Next time, please put these difficulties/issues in your question so that answers may properly address them - it would have been easier to write this in the answer instead of going back and forth in the comments like this. $\endgroup$ – KReiser Apr 15 '20 at 23:56

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