6
$\begingroup$

In a synthetic (Pappian) projective plane, one can define a conic in various clever ways not referring to coordinates. For instance, if $f$ is a projectivity from the pencil of lines through a point $A$ to the pencil of lines through another point $B$, then the locus of intersections $m \cap f(m)$ is a conic (which is nondegenerate if $f$ is not a perspectivity), and all conics are obtaned in this way.

Is there any analogous way to define curves of higher order? Say cubics, for definiteness?

Improve this question
$\endgroup$
2
  • $\begingroup$ There are many such constructions for rational (singular) curves, e.g. the strophoid, see this Wikipedia article. My intuition is that such geometric constructions will always produce rational curves, though I don't know how to justify it. $\endgroup$
    – abx
    Commented Apr 22, 2015 at 6:17
  • $\begingroup$ @abx The description of the strophoid in the Wikipedia article involves distance, which doesn't exist in a projective plane. Is there a version of it that doesn't use distances? $\endgroup$
    – Mike Shulman
    Commented Apr 22, 2015 at 7:32

1 Answer 1

Reset to default
10
$\begingroup$

It is a "classical" fact that any nonsingular plane cubic curve $C$ can be projectively generated by means of a pencil of lines and a pencil of conics.

The starting point of the construction is the observation that the lines defined by any $g_2^1$ on $C$ all pass through the same point $p$ of $C$, that following Sylvester is called the coresidual point.

Now, take four points $q_1, \ldots, q_4$ on $C$, such that any three of them are not collinear. Let $p \in C$ be the coresidual point with respect to the $g_2^1$ on $C$ cut by the pencils of conics through $q_1, \ldots, q_4$. Therefore such a pencil of conics and the pencil of lines through $p$ projectively generate $C$.

All of this is explained in the paper by N. Fraser Kötter's synthetic geometry of algebraic curves, Proceedings of the Edinburgh Mathematical Society 7, 46–61 (1888). However, the language is rather old-fashioned so the article is not easy to read nowadays.

A modern treatment can be found in Dolgachev's book Classical Algebraic Geometry, Section 3.3 (this is the googlebook link).

Improve this answer
$\endgroup$
8
  • $\begingroup$ What is a $g^1_2$? $\endgroup$
    – Mike Shulman
    Commented Apr 22, 2015 at 7:36
  • $\begingroup$ @MikeShulman - this is some particular kind of divisor, IIRC... $\endgroup$
    – Dima Pasechnik
    Commented Apr 22, 2015 at 8:09
  • $\begingroup$ It is a $1$-dimensional linear system of degree $2$ over a curve. By Riemann-Roch, since a smooth plane curve has genus $1$, every effective divisor of degree $2$ defines a $g^1_2$ on it. The classical term for such a object was "rational involution": in fact, every $g^1_2$ determines uniquely a degree $2$ morphism $C \to \mathbb{P}^1$. $\endgroup$
    – Francesco Polizzi
    Commented Apr 22, 2015 at 8:30
  • $\begingroup$ I'm assuming that the ground field is any algebraically closed field $k$ of characteristic $0$ (for instance $k= \mathbb{C})$. The chapter on curves in Hartshorne's book is a good introduction for all this stuff. $\endgroup$
    – Francesco Polizzi
    Commented Apr 22, 2015 at 8:36
  • $\begingroup$ Fraser's paper seems to be exactly what I was looking for, thanks! (It's actually easier for me to read than your description, since I'm not an algebraic geometer but I've read some synthetic projective geometry.) $\endgroup$
    – Mike Shulman
    Commented Apr 24, 2015 at 18:44

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .