The Veronese linear system on the Riemann sphere

The zero function is a polynomial. For nonzero $f\in L(D)$, [F3] writes $f=P\mathrm{/}Q$ with $P,Q$ coprime. If $Q$ were nonconstant, [F4] would give a root $a\in \mathbb{C}$; if $P(a)=0$, the same factorization result would make $z-a$ divide both $P$ and $Q$, contrary to coprimeness. Thus $P(a)\mathrm{\ne }0$ and $f$ has a finite pole at $a$, impossible for $f\in L(d[\mathrm{\infty }])$. Hence $Q$ is constant. If $P$ has degree $m$, its expression in $u=1\mathrm{/}z$ has leading term $c{u}^{-m}$, so its pole order at infinity is $m$; membership in $L(D)$ forces $m\le d$. Conversely every polynomial of degree at most $d$ has no finite poles and pole order at infinity at most $d$, so belongs to $L(D)$. The monomials are linearly independent as polynomials and span this space, proving the dimension and basis claims.

By [F6], $s_j=z^j{s}_D$ has divisor $(z^j)+D=j[0]+(d-j)[\mathrm{\infty }]$, since $z$ has a simple zero at $0$ and a simple pole at $\mathrm{\infty }$. At every finite point $s_0=s_D$ is nonzero because its divisor is $d[\mathrm{\infty }]$; at infinity $s_d$ is nonzero because its divisor is $d[0]$. Thus the basis sections have no common zero. Since $d+1\ge 2$, [F7] applies to $V=H^0(X,E)$ and gives the asserted linear-system map and pullback isomorphism.

On the source chart $z_0\mathrm{\ne }0$, the section $s_D$ is a local frame and the coefficients of $s_j$ are $z^j$, so the map has coordinates $[1:z:\cdots :z^d]$. On the chart $z_1\mathrm{\ne }0$, the coordinate is $u=z_0\mathrm{/}{z}_1=1\mathrm{/}z$ and $s_d=z^d{s}_D$ is a local frame because its divisor is $d[0]$; the coefficients of $s_0,\dots ,s_d$ in this frame are $u^d,u^{d-1},\dots ,1$. Hence the map is $[u^d:u^{d-1}:\cdots :1]$ there. On the overlap, multiplying $[1:z:\cdots :z^d]$ by $u^d$ gives the second tuple, so the formulas agree and are holomorphic in both charts; together they give the stated homogeneous formula on all of ${\mathbb{P}}^1$.

In the finite chart, the target chart coordinates are $(z,z^2,\dots ,z^d)$, whose first coordinate recovers $z$ and whose derivative has first component $1$. In the chart at infinity, the target coordinates are $(u^d,u^{d-1},\dots ,u)$, whose last coordinate recovers $u$ and whose derivative has last component $1$. The point at infinity maps to $[0:\cdots :0:1]$, outside the target chart with first coordinate nonzero, while every finite point lies in that chart. Thus the map is globally injective and has nonzero differential at every point. For any nonzero hyperplane form $L(W)={\sum }_{j=0}^d{a}_j{W}_j$, monomial independence from step 1.1 shows its pullback ${\sum }_{j=0}^d{a}_j{z}_0^{d-j}{z}_1^j$ is a nonzero homogeneous polynomial of degree $d$. If its dehomogenization on $z_0\mathrm{\ne }0$ has degree $m\le d$, [F4] gives $m$ finite roots counted with multiplicity, and in the infinity coordinate it has a zero of order $d-m$ when $m<d$; hence every hyperplane section has total multiplicity $d$. Under the hyperplane-section definition of degree, the image is the rational normal curve of degree $d$.

The chart formulas show that ${\phi }_d$ is continuous. If $C\subseteq {\mathbb{P}}^1$ is closed, [F9] makes $C$ compact; pulling any open cover of ${\phi }_d(C)$ back along ${\phi }_d$ gives an open cover of $C$, so compactness gives a finite subcover and ${\phi }_d(C)$ is compact. The target ${\mathbb{P}}^d$ is Hausdorff by [F8], so [F9] makes ${\phi }_d(C)$ closed. Hence the continuous bijection from ${\mathbb{P}}^1$ onto its image is closed and has continuous inverse. Together with the nonzero differential from step 4.1, this proves that ${\phi }_d$ is a holomorphic embedding, including the case $d=1$.