Algebraic curves–Elementary doubts on their singular points, solutions and inclined asymptotes

algebra-precalculusalgebraic-curvesalgebraic-geometrycurvesplane-curves

I have been studying algebraic curves from this book:

Rutter, J. (2000). Geometry of curves. United Kingdom: Taylor & Francis.

in which I found a few doubts

It should be remarked that I have also searched and read previous posts from this mathematics stack exchange. However, these did not help me with the questions to be presented here. Owing to the fact that algebraic curves and homogeneous polynomial equations are quite interesting topics in mathematics, I think these questions, and their possible answers, may be of great interest to those that subscribe to the mathematics stack exchange.

With that said, it is well-known that an algebraic curve is an analytical polynomial equation of the form:
$$P(x,y)=\displaystyle\sum^{n}_{i+j=0}a_{i j}x^{i}y^{j}=0$$
in which $n$ is the degree of the algebraic curve.

Based on the above, I ask:

  1. What is the best and most formal mathematical terminology for the points that satisfy the equation above?

  2. Let $(a,b)$ be a point where the above homogeneous polynomial equation is zero. Is that point necessarily a singular point?

  3. Are there any connections, results, or theorems that relate asymptotes and singular points of algebraic curves?

  4. Suppose that $x=k_{0}y+b_{0}$ is an asymptote of the above algebraic expression. Assume, further, that $P(x,y)$ is zero at
    $$s_{1}\left(x^{*},y^{*}\right)=\left(b_{0},0\right) \quad\text{and}\quad s_{2}\left(x^{*},y^{*}\right)=\left(0, -b_{0}/k_{0}\right).$$

What is the most formal mathematical terminology to refer to $s_{1}$ and $s_{2}$? That is to say, should I refer to them as roots, zeros, equilibrium points or simply solutions of $P(x,y)$?

  1. What is the maximum number of asymptotes that an algebraic curve may exhibit?

  2. Based on your wide experience, could you suggest to me references in which such definitions are discussed?

Best Answer

  1. "Points on the curve given by $f=0$" is totally correct. "The zero locus of $f$" works too. "The curve cut out by $f$" is also fine. There are also other choices.
  2. Most points are nonsingular. The precise statement is that in characteristic zero, only finitely many points on a curve are singular - to prove this, observe that a point is singular iff $f$ and all its partial derivatives vanish there. But (as long as $f$ is squarefree) $f$ and its partial derivatives are all coprime for degree reasons.
  3. Not really. Viewing asymptotes as intersections of the curve and the line at infinity, asymptotes correspond to factors of the highest degree portion of $f$. On the other hand, singularities depend on all of $f$. So given a choice of a collection of asymptotes, you can find curves with those asymptotes and different configurations of singularities (including no singularities).
  4. "Roots", "zeroes", and "solutions of $P$" are all fine. Equilibria is not a common choice unless there's something else/more going on.
  5. If the curve is cut out by $f$, then there are at most $\deg f$ asymptotes. Remember, asymptotes are given by intersections of your curve with the line at infinity - the way you get these is by homogenizing $f$ with a third variable $z$ and then setting $z=0$ and taking the (homogeneous) roots of the resulting polynomial. So there are at most $\deg f$ asymptotes.
  6. I don't have a ton of stuff on this more classical treatment of plane curves - most of the algebraic geometry I know is from a bit more of an advanced perspective: Fulton's Algebraic Curves; Arabello, Cornalba, Griffiths, and Harris' Geometry of Algebraic Curves; and then Vakil & Hartshorne both have some material on these. These are slightly little more serious texts, suitable for graduate-level study and might not be the best direct sequels to the book you mention in your post.
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