In 1919, Enriques started the study of equations defining a canonical curve of genus \(g\) in \(\mathbb{P}^{g-1}\). Soon after this was taken up by D. W. Babbage and K. Petri. Today, thanks to the work of M. Green and others [see e.g. M. L. Green, J. Differ. Geom. 19, 125- 171 (1984; Zbl 0559.14008)] this subject is still alive; in particular Green’s conjecture on canonical curves is a very stimulating problem. The work on canonical curves shows a connection between the geometry of \(C \subseteq \mathbb{P}^{g-1}\) and the minimal free resolution (m.f.r. for short) of \(C\) (that is to say, of its graded ideal). – Naturally we are led to ask which informations the m.f.r. of a curve in \(\mathbb{P}^ n\) encodes. Moreover one could try to use m.f.r. for the existence and classification problems.
In this paper we will review various connections between the m.f.r. and other geometrical properties of curves in \(\mathbb{P}^ 3\).
For the entire collection see [Zbl 0741.00049].