Synthetic projective definition of cubic curves 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?
 A: 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).
