Smooth projective curve

A smooth projective curve is a nonsingular projective variety of dimension one.

Over an algebraically closed field, a smooth projective rational curve is isomorphic to ${\mathbb{P}}^1$. Its line bundles are determined by degree, its degree-zero line bundle is trivial, and $H^1({\mathbb{P}}^1,\mathcal{O})=0$. An integral projective curve of arithmetic genus zero is automatically smooth and rational by the normalization genus formula.

At every point of a smooth algebraic curve, the local ring is a discrete valuation ring. Its valuation measures the order of vanishing of a rational function at that point.

At a point of a smooth algebraic curve, its local ring is a discrete valuation ring. A local parameter is a generator $t$ of its maximal ideal, equivalently an element of valuation one. It measures the order of a zero of a function or pole by expressing functions as powers of $t$ times units. Over the complex numbers its analytic counterpart is a local coordinate vanishing simply at the point. For an elliptic curve in a Weierstrass equation of an elliptic curve, $t=-x/y$ is the standard parameter at the identity.

Every rational map from a smooth projective curve to a projective variety extends uniquely to a morphism. At a missing point, the discrete valuation ring of the curve lets one divide homogeneous coordinates by their smallest valuation, leaving regular coordinates of which at least one is a unit.

Let $\nu :\tilde{C}\to C$ be the normalization of a nodal curve. Its rational inverse $C⇢\tilde{C}$ cannot extend over the node: the two branches give two distinct points of $\tilde{C}$, whereas a morphism can assign only one image to the node.