Given a homology class $\alpha \in H_k(X,\mathbb{Z})$ on a variety $X$, is there a way to determine if there exists an irreducible subvariety $Y \subset X$ that has that class, i.e. $[Y] = \alpha$?
For practical reasons, assume that $X$ is a toric variety, so that one has a good handle on the cohomology ring of $X$. For example, if $X = \mathbb{P}^2$, with $H \in H_2(\mathbb{P}^2,\mathbb{Z})$ the hyperplane class, and $\alpha = 2H$, then a possible $Y$ would be the curve defined by $x y -z^2 =0$, where $x,y,z$ are the usual projective coordinates. Could this information - the existence of such a curve - solely be deduced from the properties (degree, intersection numbers, ect.) of $\alpha$?
To narrow down the problem further: Say I work in a smooth, complete two-dimensional toric variety $X$, where I know that $Pic(X)$ is generated by the toric divisors. These are the classes the zero loci $x_i=0$, where $x_i$ are the coordinates associated to the vertices of the fan of X. Now if I take an linear combination $\alpha = \sum_i \lambda_i [x_i]$, is there a irreducible subvariety in this class?
Many thanks in advance!
EDIT: Are there also necessary conditions for the existence of irreducible representatives? I.e. can I say what the circumstances are under which there is never a irreducible variety in a given divisor class.