projective non-singular curve

Question

I am working on algebraic curves at the moment and I can not find a proper definition of the projective non-singular curves. My goal is understand that the category of non-singular projective curves is equivalent to the finite generated field extension of k of transendence degree 1. Now an abstract non singular curves is just a open subset over $C_k$ where this is just the set of all dvr.

But what is a projective non-singular curve. I am glad for any hints.

Answer 1

1. variety: integral scheme of finite type over a field. (This may vary a lot in literature: one can add "separated", remove "integral", add "reduced" again...).
2. curve: one-dimensional variety.
3. projective (over a field): closed subvariety of some projective space (over that field).
4. nonsingular: all local rings at closed points are regular local rings.

Hence a projective nonsingular curve is a one-dimensional projective variety all of whose local rings (at closed points) are discrete valuation rings (as DVR means: regular local ring of dimension one).

The equivalence of categories you are working with is provided by the function field functor: $$C\mapsto K(C)=\{\textrm{rational functions on }C\}.$$ It is important to note that for $K(-)$ to be an equivalence, one has to consider the category of curves with arrows the dominant morphisms of curves. In addition, a third category is equivalent to the tho above: that of curves, with dominant rational maps.