= Solution
For a <divisor on an algebraic curve> $D$ on a smooth projective curve $X$ of genus $g$, the <Riemann-Roch theorem> states
$$
\ell(D)-\ell(K_X-D)=\deg D+1-g,
$$
where $K_X$ is a <canonical divisor>. Taking $D=0$ gives $1-\ell(K_X)=1-g$, so $\ell(K_X)=g$. Taking $D=K_X$ then gives
$$
g-1=\deg K_X+1-g,
\qquad
\boxed{\deg K_X=2g-2}.
$$
To obtain a uniform projective embedding, choose a divisor $D$ of degree $2g+1$. Since every divisor appearing below has degree greater than $2g-2$, Riemann--Roch gives
$$
\ell(D)=g+2,
\quad
\ell(D-P)=g+1,
\quad
\ell(D-P-Q)=g,
\quad
\ell(D-2P)=g.
$$
The first two equalities show that the <complete linear system of a divisor> $|D|$ has no base point. The strict drops in the last two comparisons show respectively that its sections separate distinct points $P,Q$ and tangent directions at $P$. Thus $D$ is a <very ample divisor>, in accordance with the general fact that a <high-degree divisor is very ample on a smooth projective curve>, and its sections define a closed embedding
$$
\boxed{X\hookrightarrow\mathbb P^{\ell(D)-1}=\mathbb P^{g+1}.}
$$
The ambient dimension therefore depends only on $g$.
The <Riemann-Hurwitz formula> for a nonconstant morphism $h:C_1\to C_2$ of degree $n$ is
$$
2g(C_1)-2=n\bigl(2g(C_2)-2\bigr)
+\sum_{P\in C_1}(e_P-1).
$$
Choose a <smooth plane quartic> $C$, so $g(C)=3$, and form the <product of projective varieties> $S=C\times C$. This is a smooth projective variety of dimension two. If $B\subset S$ is an irreducible curve, pass to its <normalization of an algebraic curve>[normalization] $\widetilde B$. At least one coordinate projection $\widetilde B\to C$ is nonconstant, since otherwise $B$ would be a point. For that projection, Riemann--Hurwitz gives
$$
2g(\widetilde B)-2
\geq n(2\cdot3-2)\geq4,
$$
so the <geometric genus> of $B$ is at least three. Hence $S$ is the required surface; this is the <product surface without low-genus curves> construction.
Finally let $X\subset\mathbb P^2$ be a smooth plane curve of degree $d$ and let $p\notin X$. After a projective change of coordinates, take $p=[0:0:1]$. Projection away from $p$ is
$$
[x:y:z]\longmapsto[x:y].
$$
Its two homogeneous coordinate functions cannot vanish simultaneously on $X$, because their common zero in $\mathbb P^2$ is $p$. The criterion for a <morphism of algebraic varieties> therefore shows that the restriction
$$
\pi:X\longrightarrow\mathbb P^1
$$
is a morphism. A fibre is the intersection with a line through $p$, and a general such line meets $X$ in $d$ points counted with multiplicity. Thus the <projection of a plane curve from an exterior point> has degree $d$.
By the <genus of a smooth plane curve>, $g(X)=(d-1)(d-2)/2$. Applying Riemann--Hurwitz to $\pi$ and its <ramification divisor> $R_\pi$ gives
$$
\deg R_\pi
=2g(X)-2+2d
=(d-1)(d-2)-2+2d
=d(d-1).
$$
Every ramification point contributes at least one to this degree, so
$$
\boxed{\#\{P\in X:\pi\text{ ramifies at }P\}\leq d(d-1)}.
$$
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