This is related to Ueno's Algebraic Geometry 3, Example 7.33 of Chpt 7, Sec 2.
Suppose $\operatorname{char}(k)\neq 2$ and $f(x)$ has no repeated roots in algebraic closure. Consider $y^2=f(x)$ with $\deg(f)\geq 3$. However, naive projectivization in $P^2_k$ might be singular due to point at infinity. Denote the scheme $X=\operatorname{Spec}(k[x,y]/(y^2-f(x)))$.
The book goes through construction of a line bundle $L$ as a scheme over $P^1_k$ s.t. $\Gamma(P^1_k,L)\cong \Gamma(P^1_k,O(\deg(f)))$ where first $\Gamma(P^1_k,-)$ is global section of scheme $L\to P^1_k$. The construction of $L$ is roughly giving transition maps with appropriate degree for twisting. In particular, $X$ can be realized as a section over open subset. $X$'s closure is $\tilde{X}$ in $L$.
Here is detailed $L$ construction. $P^1_k=\operatorname{Proj}(k[x_0,x_1])$. Then $U=D(x_0),V=D(x_1)$ covers $P^1_k$. Let $x=\frac{x_1}{x_0}$ and $y=\frac{x_0}{x_1}=\frac{1}{x}$. Define $U'=k[x,u]$ and $V'=k[y,v]$ as trivialization of $L$ over $U$ and $V$ respectively.(i.e. $U'\to U$ and $V'\to V$ are canonical projection of $U\times_{Spec(Z)} A^1_k\to U$ and $V\times_{Spec(Z)} A^1_k\to V$.) It suffices to define the isomorphism to identify $U',V'$ over $U\cap V$. Consider $k[x,\frac{1}{x},u]\to k[y,\frac{1}{y},v]$ by $x\to\frac{1}{y}$ and $u\to\frac{v}{y^m}$. This identification gives identification. Then this describes $L\to P^1_k$ by gluing the corresponding projection maps.
$\textbf{Q:}$ Is this $\tilde{X}$ blow up of $X$ without acknowledging this fact? Naively taking projective closure of $X$ in $P^2_k$ will give rise to a singular point at $\infty$. Thus it is necessary to blow up at that point. If so, how should I realize such a thing? Furthermore, can every blow up be realized in this fashion? What is the most general construction here?
$\textbf{Q:}$ Is every section of $O(m)$ as section of $L$ over $P^1_k$ non singular?