Let $X$ be a smooth projective and irreducible curve over a field $k$. Further, define $$ X_d = \{ \text{ Effective Cartier divisors of degree } d \text{ on } X \;\} $$ and $$ W_d = \{ \text{ Line bundles of degree } d \text{ on } X \;\}. $$ What is the best way to give the above sets a scheme structure and show that those schemes are irreducible?
I'm not sure, intuitively, if the irreducibility holds in general or if we need further assumptions on $X$and/or $k$.
PS: A way to give a scheme structure to $X_d$ is the following: Assume $X$ is a scheme over another scheme $S$ and consider the functor $$ Div^d_{X/S}: Sch_S \to Set, \quad T\mapsto \{ \text{ Relative eff Car divisors of deg } d \text{ on } X_T/T \; \} $$ and assume that it is representable (this is true for $X$ curve as above). Then one defines $X_d$ as the representing $S$-scheme.
We can define $W_d$ similarly with an appropriate functor $\;Sch_S \to Set$.
Effective divisors of degree $d$ can be seen as the symmetric product $\mbox{Sym}^d(X):=X^d/S_d$ where $S_d$ is the symmetric group on $d$ letters. $X^d$ is irreducible since $X$ is, and so $X^d$ and $\mbox{Sym}^d(X)$ receive a natural structure of variety.
As far as $W_d$, identify $JX$, the Jacobian of $X$, with $\mbox{Pic}^0(X)$. Let $L_d$ be a line bundle of degree $d$ on $X$, and define the map $\mbox{Pic}^0(X)\to\mbox{Pic}^d(X)$ where $L\mapsto L\otimes L_d$ where $\mbox{Pic}^d(X)$ denotes the set of line bundles of degree $d$ on $X$. This is a (non-canonical) isomorphism, and since $JX$ is irreducible and has the structure of variety, we get the same for $\mbox{Pic}^d(X)$