= Solution
Use <projectivization by quotients>, and set
$$
E=\bigoplus_{i=1}^n\mathcal O_{\mathbb P^1}(a_i),
\qquad F=\mathbb P_{\mathbb P^1}(E),
\qquad \pi:F\to\mathbb P^1.
$$
The fibre over $t$ parametrizes one-dimensional quotients of $E_t$, so it is $\mathbb P^{n-1}$. The standard <line bundles> are
$$
L=\pi^*\mathcal O_{\mathbb P^1}(1),\qquad M=\mathcal O_F(1).
$$
Thus $L$ is the <fiber class of a rational normal scroll> of the ruling and $M$ is the <universal quotient line bundle> of $\pi^*E$. In additive divisor notation we also write $L,M$ for their first <Chern classes>. For $n\ge2$ these generate the <Picard group>; for $n=1$, $F=\mathbb P^1$ and $M=a_1L$. This fixes the sign convention for the <rational normal scroll>.
Put $s=\sum_i a_i$. The <projective bundle formula for Chow groups> gives
$$
A^*(F)=\mathbb Z[L,M]/(L^2,\ M^n-sLM^{n-1}).
$$
Here the first relation comes from the one-dimensional base, and the second is the projective-bundle relation with $c_1(E)=sL$ and higher <Chern classes> zero on the base. Integration over a fibre gives $LM^{n-1}=1$. Therefore all top-degree mixed <intersection numbers> are
$$
\boxed{M^n=s,\qquad LM^{n-1}=1,\qquad
L^jM^{n-j}=0\quad(2\le j\le n).}
$$
In particular, for integers $u,v$,
$$
(uM+vL)^n=u^ns+nu^{n-1}v.
$$
For the <canonical divisor of a smooth variety>, the relative Euler sequence
$$
0\to\mathcal O_F\to\pi^*E^*\otimes\mathcal O_F(M)
\to T_{F/\mathbb P^1}\to0
$$
has determinant $\mathcal O_F(nM-sL)$. Combining its dual determinant with $K_{\mathbb P^1}=-2L$ gives
$$
\boxed{K_F=-nM+(s-2)L.}
$$
A common twist does not change the abstract <projective bundle>. Choose an integer $b$ such that $d_i=a_i+b\ge1$ for every $i$. The complete <linear system of divisors> $|M+bL|$ has
$$
h^0(F,M+bL)=h^0(\mathbb P^1,E(b))
=\sum_i(d_i+1)=s+nb+n.
$$
Its map to $\mathbb P^N$, where $N=s+nb+n-1$, has coordinate blocks
$$
\lambda_i s_0^{d_i},\
\lambda_i s_0^{d_i-1}t_0,\ \ldots,\
\lambda_i t_0^{d_i}\qquad(1\le i\le n).
$$
Here $[s_0:t_0]$ is a point of the base and the $\lambda_i$ are quotient coordinates in a local trivialization of $E(b)$. These expressions have the correct transition factors and define the global map. Any nonzero block recovers the base point through its degree-$d_i$ <rational normal curve>. Once that point is known, the blocks recover the projective fibre coordinates. On the open where the first coordinate of a nonzero block is nonzero, the ratio of the next coordinate to the first recovers $t_0/s_0$; the analogous last-coordinate chart recovers $s_0/t_0$. The remaining coordinate ratios then recover the fibre coordinates regularly. Thus the map identifies $F$ with its closed image and is a <projective embedding>, not merely a pointwise <injection>.
The hyperplane class on this image is $M+bL$, so
$$
\deg F=(M+bL)^n=s+nb,\qquad
\operatorname{codim}F=N-n=s+nb-1.
$$
Consequently every such natural embedding has
$$
\boxed{\deg F=\operatorname{codim}F+1.}
$$
The tautological choice $b=0$ is an embedding when all $a_i>0$. If the $a_i$ are only nonnegative, $|M|$ is <basepoint-free> but can contract the projective subbundle arising from zero summands; one must distinguish that <scroll> image from the smooth abstract <scroll>.
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