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Let $E$ be a genus one curve over an algebraically closed field and choose $O\in E$. The mapis an isomorphism. For a degree-zero divisor $D$, Riemann--Roch gives a nonzero section of $D+O$, so $D\sim P-O$ for some point $P$. If $P-O\sim P^{\prime }-O$ with $P\ne P^{\prime }$, a rational function with divisor $P-P^{\prime }$ would give a degree-one map to the projective line, contradicting that $E$ has genus one. The Abel--Jacobi map transports addition in the divisor class group to the elliptic-curve group law.

The complex projective plane is the two-dimensional Complex projective space. Its second homology is generated by the class $H$ of a projective line, with $H^2=1$.

Every degree-zero divisor on the projective line is principal. Indeed, if $D={\sum }_i{a}_i{p}_i$ and $L_i$ is a homogeneous linear form vanishing at $p_i$, then ${\sum }_i{a}_i=0$ makesa degree-zero rational function with $(f)=D$. Consequently, two divisors on ${\mathbb{P}}^1$ are linearly equivalent exactly when they have the same degree.

Projection from a point of a smooth plane quartic gives a degree-three map to the projective line. A degree-two map would make the curve hyperelliptic, but its canonical map is its plane embedding, whereas a hyperelliptic canonical map is not an embedding. Its gonality is therefore three.

If a projective line $H$ does not contain the projective plane curve $X$, its hyperplane-section divisor iswhere $I_p(X,H)$ is the intersection multiplicity. The Bézout theorem gives $\mathrm{deg}[X\cap H]=(\mathrm{deg}X)(\mathrm{deg}H)=\mathrm{deg}X$.

On the projective line, an infinite direct sum of copies of $\mathcal{O}(-1)$ is a quasi-coherent sheaf but not a coherent sheaf: at any point its residue-vector-space dimension is infinite. Its global sections vanish because each summand has no sections, and sections commute with direct sums for the finite two-chart equalizer. Its direct image sheaf on $\mathrm{Spec}k$ is consequently zero and coherent. This shows that a nonaffine direct image can lose information needed to detect coherence.

Over an algebraically closed field of characteristic different from two, a degree-two map from a smooth genus one curve to the projective line has a deck involution. After choosing an origin, all its fiber divisors on an algebraic curve are linearly equivalent and have the same group sum $s$, by the Abel-Jacobi map of a genus-one curve. The involution is consequently $p\mapsto s-p$, including its fixed ramification points. Composing two such involutions gives a translation on an elliptic curve.

An elliptic curve in characteristic zero admits no nonconstant rational map of projective varieties from the projective line. Over $\mathbb{C}$, write its equation as $y^2={\prod }_{j=1}^3(x-e_j)$ with distinct $e_j$. Substituting $x=u/v$ with coprime polynomials gives $(y{v}^2{)}^2=v{\prod }_j(u-e_{j}v)$. These four pairwise coprime factors must all be squares. The polynomial pencil with four square members forces $u,v$ constant, ruling out a nonconstant rational parametrization of an algebraic curve.