Projective curve

A projective curve is a projective algebraic curve. A projective curve is a one-dimensional projective variety.

A rational normal curve in ${\mathbb{P}}^n$ is the image of the degree-$n$ Veronese map from ${\mathbb{P}}^1$. After projective changes of coordinates it has parametrization

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

A nonconstant rational function on a smooth projective curve extends to a finite morphism to the projective line. The inverse images of its two standard affine charts are affine varieties covering the curve. The resulting Čech cochain complex has only degrees zero and one, so every coherent sheaf has zero sheaf cohomology in degrees at least two.

For a smooth projective curve with $H^0(C,{\mathcal{O}}_C)=k$, its genus is ${\dim }_k{H}^1(C,{\mathcal{O}}_C)$, equivalently ${\dim }_k{H}^0(C,{\Omega }_C^1)$ by Serre duality. It equals the arithmetic genus $1-\chi ({\mathcal{O}}_C)$. Without the condition on constants, the general Riemann-Roch theorem instead uses $\chi ({\mathcal{O}}_C)=h^0({\mathcal{O}}_C)-h^1({\mathcal{O}}_C)$.

Over an algebraically closed field, choose a point $P$ on the smooth projective curve. The Riemann-Roch theorem gives $\dim L(P)=2$: the complementary canonical divisor term has negative degree. A nonconstant function in $L(P)$ has exactly one simple pole, so it defines a degree-one morphism of algebraic curves to the projective line. A degree-one map between smooth projective curves is an isomorphism. The existence of a rational point is essential over a field which is not algebraically closed.

Over the algebraic closure of ${\mathbb{F}}_3$, the displayed cubic is smooth. Its three partial derivatives are $Z^2$, $2YZ$, and $Y^2+2XZ$; their simultaneous vanishing would give $Z=Y=0$, and the equation would then give $X=0$, impossible in projective space. Its affine equation $y^2=x^3-x$ is irreducible because the odd-degree polynomial on the right is not a square in the rational function field.

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.