A map from an algebraic variety $X$ to a projective space is the same thing as a globally generated line bundle on $X$. What geometric object on $X$ corresponds to a map to a *weighted* projective space?



Maps to projective space == line bundles; what do maps to weighted projective space correspond to?
Let me expand my comment in a short answer.


One of the most common way to build a rational map $f \colon X \dashrightarrow \mathbb{P}(a\_1, \ldots, a\_n)$ is to consider a line bundle $\mathscr{L}$ on $X$ together with sections $$\sigma\_1 \in H^0(X, \, \mathscr{L}^{a\_1}), \quad \sigma\_2 \in H^0(X, \, \mathscr{L}^{a\_2}), \ldots, \sigma\_n \in H^0(X, \, \mathscr{L}^{a\_n}),$$
and to define $f(x) := [\sigma\_1(x): \ldots : \sigma\_n(x)].$ 


Clearly, $f$ is a morphism if and only if the intersection of the zero loci of the sections $\sigma\_i$ is empty.


The basic example is given by hyperelliptic curves. Let $X$ be a smooth hyperelliptic curve of genus $g$ and take as $\mathscr{L}$ the hyperelliptic involution, i.e. the unique $g^1\_2$ of $X$. Then it is no difficult to show that there exist $x\_0, \, x\_1 \in H^0(X, \, \mathscr{L})$ and $y \in H^0(X, \mathscr{L}^{g+1})$ that satisfy a polynomial relation of type $$y^2 = F\_{2g+2}(x\_0, \, x\_1), \quad (\sharp)$$
where $F\_{2g+2}$ is a homogeneous form of degree $2g+2$.


This precisely means that we can write $X$ as a double cover of $\mathbb{P}^1$, branched at $2g+2$ points. Trying to homogenize the equation $(\sharp)$ in the usual way introduces a complicate singularity at infinity. Instead, it is more convenient to see $(\sharp)$ as the *global* equation of $X$ into the weighted projective plane $\mathbb{P}(1, \, 1,\, g+1)$.


In fact, it is easy to see that the map $f \colon X \longrightarrow \mathbb{P}(1, \, 1, \, g+1)$ defined by the sections $x\_0, \, x\_1, \, y$ is an embedding (notice that the image does not pass through the singular point $[0: 0: 1]$ of the weighted projective plane). 


