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A rational parametrization expresses the coordinates of an algebraic curve as rational functions of a parameter, with a rational inverse on a dense open subset. It identifies the function field with $K(t)$ and the smooth projective normalization of an algebraic curve with the projective line. A nonconstant parametrizing map without an inverse is a weaker notion; even this cannot exist for an elliptic curve in characteristic zero.

This smooth projective plane curve has no real rational points: a sum of three real squares is zero only when all three coordinates vanish, which is forbidden in projective space. It is therefore not isomorphic over $\mathbb{R}$ to the projective line, which has rational points. Over $\mathbb{C}$ it becomes a nonsingular conic isomorphic to a projective line. Thus the ground field changes whether a geometric genus-zero curve admits a projective-line parametrization.

Let the base be a smooth algebraic curve, the total surface be smooth, and the field be algebraically closed of characteristic different from two. At a rank-two singular fibre with kernel vector $v$, the fibre derivatives vanish at $[v]$. Smoothness requires $v^{\mathsf{T}}{M}^{\prime }v\ne 0$. The adjugate matrix is a nonzero multiple of $v{v}^{\mathsf{T}}$, so the determinant has a simple zero. At rank at most one, the projective kernel contains a line and the quadratic $v^{\mathsf{T}}{M}^{\prime }v$ has a zero on it; that point would make the total surface singular. Thus every singular fibre is a pair of distinct projective lines and its discriminant zero is simple.

Use the rational principal-parts resolution on an algebraic curve to represent a class by finitely many rational differential principal parts. Sum their algebraic residues of a rational differential. The sum of residues of a global rational differential is zero: trace to the projective line, where partial fractions prove the assertion. Hence the sum descends to cohomology. Pairing an invertible sheaf with its differential-valued dual gives the perfect Serre duality pairing. Since the global regular functions on an integral projective curve over an algebraically closed field are $k$, evaluation at the constant function $1$ identifies this trace canonically with an isomorphism to $k$.

Under $Q\cong {\mathbb{P}}^1\times {\mathbb{P}}^1$, the fibres of the two coordinate projections are the two rulings of the smooth quadric surface. Two distinct fibres in one ruling are disjoint projective lines, while one fibre from each ruling meets in one point.

Take rays $(i,1)$ and cones generated by consecutive rays in ${\mathbb{Z}}^2$. Their dual generators $(1,-i)$ and $(-1,i+1)$ give $x_i=t{q}^{-i}$, $y_i=t^{-1}{q}^{i+1}$. Adjacent charts glue by $x_{i+1}=y_i^{-1}$, $y_{i+1}=x_i{y}_i^2$. The fibre $q=0$ is a chain of projective lines: each component has two affine-line charts joined by reciprocal coordinates, and consecutive components meet at one node. Inverting $q$ identifies all charts with the multiplicative group scheme over $\mathbb{Z}[q,q^{-1}]$. The local node algebra is free over $\mathbb{Z}[q]$, with basis $1,x_i^a,y_i^b$ for $a,b\ge 1$, so the morphism is flat.

// Target: toric-geometry.bigb

For any field $F$, a projective linear group element is uniquely determined by its images of three distinct points of the projective line, and any ordered distinct target triple is attainable. Given representatives $v_z,v_x$ forming a basis and $v_y=A{v}_z+B{v}_x$, the matrix with columns $A{v}_z,B{v}_x$ sends $(0,1,\infty )$ to $(x,y,z)$. Distinctness forces $A,B\ne 0$. The matrices realizing one projective map differ by a nonzero scalar; over ${\mathbb{F}}_p$ there are $p-1$ such matrices.

Exactly one element sends each ordered triple of distinct points to each other such triple. On a projective line, representatives of the first two target points form a basis; scaling them so their sum represents the third point gives the unique projective transformation.

A matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ acts on the projective line by the Möbius transformation $x\mapsto (ax+b)/(cx+d)$, with the pole sent to infinity and infinity sent to $a/c$ when $c\ne 0$. The action is transitive because $\left(\begin{array}{cc}t& -1\\ 1& 0\end{array}\right)$ sends infinity to any finite $t$. The stabilizer subgroup of infinity consists of $\left(\begin{array}{cc}a& b\\ 0& a^{-1}\end{array}\right)$ with $a\ne 0$. The orbit-stabilizer theorem gives $|{\mathrm{SL}}_2({\mathbb{F}}_q)|=q(q^2-1)$. Its kernel is the scalar matrices of determinant one: a transformation fixing infinity has $c=0$, fixing zero then forces $b=0$, and fixing one gives $a=d$. Thus the kernel is $\{I,-I\}$, with these matrices coinciding in characteristic two.

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.