TITLE: Space Curves as Determinantal Varieties
QUESTION [11 upvotes]: I read in a paper of Goryunov
(`Functions on space curves', 
Journal of The London Mathematical Society, vol. 61 (2000), 807-822;
available on his home page)
that every space curve can be defined as the vanishing
of the N minors of some  N by N+1 matrix with entries functions of x, y, z.
How does one prove this? 
(I can do it for the  rational normal curve, thanks to Harris's book.)

REPLY [19 votes]: Let $I\subset R = k[x,y,z]$ be the defining ideal of  your curve. Then $R/I$ has dimension one and no embedded components, so has projective dimension $2$ by the Auslander-Buchsbaum formula. Therefore $I$ itself has projective dimension $1$, and so can be fit into a short exact sequence:
$$0 \to F \to G \to I \to 0 $$
with $F,G$ free (a minor point: one needs that projective modules are free here, it is easy if you assume $I \subset (x,y,z) $, since you may as well look at the local ring at the origin). If $N=\text{rank} F$, then $\text{rank} G=N+1$, and the matrix representing the map from $F$ to $G$ is what you want. This is known as the Hilbert-Burch theorem, and details can be found in Chapter 20 of Eisenbud's book "Commutative Algebra with a view..."