Divisors and complete linear systems on the projective line

Divisors of polynomials. By [F1] a monic irreducible $g\in k[t]$ of degree $e$ has $\mathrm{div}(g)=[V(g)]-e[\mathrm{\infty }]$, in particular $\mathrm{div}(t)=[V(t)]-[\mathrm{\infty }]$; factor a nonzero $p\in k[t]$ as $p=c{\prod }_i{g}_i^{e_i}$ with monic irreducibles $g_i$ and $c\in k^{\times }$ [F8]. Additivity of $\mathrm{ord}$ and $\mathrm{div}(fg)=\mathrm{div}(f)+\mathrm{div}(g)$ [F2] give $\mathrm{div}(p)={\sum }_i{e}_i[V(g_i)]-({\sum }_i{e}_i\mathrm{deg}{g}_i)[\mathrm{\infty }]=Z(p)-(\mathrm{deg}p)[\mathrm{\infty }],$ where $Z(p)={\sum }_i{e}_i[V(g_i)]$ is effective with ${\mathrm{deg}}_{k}Z(p)={\sum }_i{e}_i\mathrm{deg}(g_i)=\mathrm{deg}p$ and is supported away from $\mathrm{\infty }$; for a constant $p=c$ this reads $\mathrm{div}(c)=0$.

The Riemann-Roch spaces of $d[\mathrm{\infty }]$. Let $f=P\mathrm{/}Q\mathrm{\ne }0$ with coprime $P,Q\in k[t]$ [F8]; by step 1.1, $\mathrm{div}(f)=Z(P)-Z(Q)+(\mathrm{deg}Q-\mathrm{deg}P)[\mathrm{\infty }]$, and $f\in L(d[\mathrm{\infty }])$ means $\mathrm{div}(f)+d[\mathrm{\infty }]\ge 0$ [F3]. If $Q=c{t}^m$ then $\mathrm{div}(f)+d[\mathrm{\infty }]=Z(P)-m[0]+(m-\mathrm{deg}P+d)[\mathrm{\infty }]$, where $0=V(t)$ is the origin and $t\nmid P$ by coprimality; effectivity forces $m=0$, so $f=P\in k[t]$ and $\mathrm{deg}P\le d$. If $Q$ has an irreducible factor $g\mathrm{\ne }t$, then $\mathrm{div}(f)$ has the strictly negative coefficient $-e$ at $V(g)\mathrm{\ne }\mathrm{\infty }$, so $\mathrm{div}(f)+d[\mathrm{\infty }]$ is not effective. Hence $L(d[\mathrm{\infty }])=\{P\in k[t]:P=0\text{ or }\mathrm{deg}P\le d\},$ the polynomials of degree at most $d$ together with $0$: a $k$-vector space with basis $1,t,\dots ,t^d$ for $d\ge 0$, of dimension $d+1$, and the zero space for $d<0$. Moreover ${\mathrm{deg}}_k(d[\mathrm{\infty }])=d\cdot [\kappa (\mathrm{\infty }):k]=d$ because $\kappa (\mathrm{\infty })=k$ [F2].

The associated morphism is the Veronese embedding. Let $d\ge 1$ and let $V=L(d[\mathrm{\infty }])$, a $(d+1)$-dimensional subspace of $L(d[\mathrm{\infty }])$ [F3]. No point of ${\mathbb{P}}_k^1$ is a base point of $V$: at a closed point $x\mathrm{\ne }\mathrm{\infty }$ the constant function $1$ does not vanish, since $\mathrm{div}(1)+d[\mathrm{\infty }]=d[\mathrm{\infty }]$ is supported at $\mathrm{\infty }$, while at $x=\mathrm{\infty }$ the polynomial $t^d$ does not vanish, since $\mathrm{div}(t^d)+d[\mathrm{\infty }]=d[0]$ is supported at the origin $0$ [F4, step 1.1]. So $V$ is base-point-free, and [F5] attaches to it a $k$-morphism ${\phi }_V:{\mathbb{P}}_k^1\to {\mathbb{P}}_k^d$ whose pullback of the coordinate sections is the basis $1,t,\dots ,t^d$ of $V$ and whose hyperplane pullbacks are the members of $\mathrm{\mid }d[\mathrm{\infty }]\mathrm{\mid }$; the same morphism is obtained from the generating sections $1,t,\dots ,t^d$ of the invertible sheaf $\mathcal{O}(d[\mathrm{\infty }])$ by the universal property [F4]. On the chart $x_0\mathrm{\ne }0$ of ${\mathbb{P}}_k^d$ the chart formula of [F4] gives $x_j^{(0)}\circ {\phi }_V=t^j$ on the open where the section $1$ is invertible, which is ${\mathbb{P}}_k^1\setminus \{\mathrm{\infty }\}$; hence ${\phi }_V([1:t])=[1:t:t^2:\cdots :t^d]$ for every $t\mathrm{\ne }\mathrm{\infty }$. The degree-$d$ Veronese map ${\nu }_{1,d}$ of [F6] reads $[x_0:x_1]\mapsto [x_0^d:x_0^{d-1}{x}_1:\cdots :x_1^d]=[1:t:\cdots :t^d]$ on the same chart $\{x_0\mathrm{\ne }0\}={\mathbb{P}}_k^1\setminus \{\mathrm{\infty }\}$. Since ${\mathbb{P}}_k^1$ is reduced and ${\mathbb{P}}_k^d$ is separated over $k$, [F7] gives ${\phi }_V={\nu }_{1,d}$; by [F6] this morphism is a closed immersion, the degree-$d$ Veronese embedding.

The complete linear system of $d[\mathrm{\infty }]$. For $d\ge 0$ let $D^{\mathrm{\prime }}$ be an effective divisor of degree $d$ on ${\mathbb{P}}_k^1$; by [F1] every divisor on ${\mathbb{P}}_k^1$ is linearly equivalent to ${\mathrm{deg}}_k(D^{\mathrm{\prime }})[\mathrm{\infty }]=d[\mathrm{\infty }]$, so $D^{\mathrm{\prime }}\in \mathrm{\mid }d[\mathrm{\infty }]\mathrm{\mid }$ [F3]. Conversely every $D^{\mathrm{\prime }}\in \mathrm{\mid }d[\mathrm{\infty }]\mathrm{\mid }$ is effective by definition and has ${\mathrm{deg}}_k{D}^{\mathrm{\prime }}={\mathrm{deg}}_k(d[\mathrm{\infty }])=d$, since linearly equivalent divisors have equal degree [F2]. Hence $\mathrm{\mid }d[\mathrm{\infty }]\mathrm{\mid }=\{D^{\mathrm{\prime }}\text{ effective on }{\mathbb{P}}_k^1:{\mathrm{deg}}_k{D}^{\mathrm{\prime }}=d\}$ for $d\ge 0$, a set parametrised by $P(L(d[\mathrm{\infty }]))$ and hence by the projective space ${\mathbb{P}}^d$ of $k$-lines in the $(d+1)$-dimensional space $L(d[\mathrm{\infty }])$; for $d<0$ the space $L(d[\mathrm{\infty }])$ is zero by step 2.1, so $\mathrm{\mid }d[\mathrm{\infty }]\mathrm{\mid }=\mathrm{\varnothing }$.

Conclusion. For every nonzero polynomial $p(t)$ of degree $m$ one has $\mathrm{div}(p)=Z(p)-m[\mathrm{\infty }]$ with $Z(p)$ effective of degree $m$ (step 1.1); the space $L(d[\mathrm{\infty }])$ is exactly the space of polynomials of degree at most $d$ together with $0$, of dimension $d+1$ for $d\ge 0$ and zero for $d<0$ (step 2.1); the complete linear system $\mathrm{\mid }d[\mathrm{\infty }]\mathrm{\mid }$ is the set of effective divisors of degree $d$ for $d\ge 0$ and is empty for $d<0$ (step 3.1). Its set of members is the set of $k$-lines in $L(d[\mathrm{\infty }])$, identified for $d\ge 0$ with the $k$-rational points of its projective scheme of lines. For $d\ge 1$ the associated morphism is the degree-$d$ Veronese closed immersion; for $d=0$ it is the constant map to ${\mathbb{P}}_k^0$; and for $d<0$ there is no associated projective morphism. Choice enters through the current divisor, cohomology, Cartier/Weil and projective-space suppliers [F1], [F4], [F5], [F7], [F8] and [F9]; AC supplies DC for the Cartier-to-Weil interface. No further selection occurs.