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?



Synthetic projective definition of cubic curves
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](http://journals.cambridge.org/action/displayFulltext?type=1&fid=3146652&jid=PEM&volumeId=7&issueId=-1&aid=3131388&bodyId=&membershipNumber=&societyETOCSession=)*, 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](https://books.google.it/books?id=g9GLXmR9qg0C&pg=PA133&lpg=PA133&dq=projective%20generation%20of%20cubics&source=bl&ots=9yrG2MNEYv&sig=Xn9pePQ98v8ogYUuPbV7IAGcPEI&hl=it&sa=X&ei=MkA3VYSfFpHoaMO_gcAN&ved=0CC4Q6AEwAQ#v=onepage&q=projective%20generation%20of%20cubics&f=false) is the googlebook link).


