I'm interested in deformations of subvarieties of a toric variety $X$. Suppose we know a subvariety $V$ in the Chow group of $X$, for example, $V$ is a linear combination of powers of hypersurfaces. Can we deduce out of here some meaningful information about dimension of the deformation space of $V$?

For example, in the case of curves on a toric surface we can find the dimension of the deformation space of $V$ as its self-intersection.

I'm interested in deformations of subvarieties of a toric variety $X$. Suppose we know a subvariety $V$ in the Chow group of $X$, for example, $V$ is a linear combination of powers of hypersurfaces. Can we deduce out of here some meaningful information about dimension of the deformation space of $V$?

For example, in the case of curves on a toric surface we can find the dimension of the deformation space of $V$ as its self-intersection. Similar arguments work for any hypersurface: its deformation space is encoded by its $n+1$-th power, where $n$ is its dimension.