Let $\mathbb{P}^n$ be the complex projective $n$-space and let $\mathcal{H}$ be the dual bundle of the tautologic line bundle over $ \mathbb{P}^n$.
I have a feeling that the author of a paper I'm reading is calling a section of $\mathcal{H}^{\otimes r}$ a "homogeneous polynomial of degree $r$ in $n+1$ variables". I can (kind of) roughly see why one might conceive of a section of this bundle as such.
My question is: is there an exact correspondence between homogeneous polynomials of degree $r$ in $n+1$ variables and sections of $\mathcal{H}^{\otimes r}$, or is it just convenient and intuitive shorthand, or neither?