Following the work of Miranda for triple covers, given a covering map $f\colon X\rightarrow Y$, I will show how to determine the structure as an algebra of the $\mathbb{O}_Y$ module $f_*\mathbb{O}_X$. When applied to Gorenstein covering maps this method brings the structure theorem of Casnati and Ekedahl to new light and can be used to describe a family of codimension $4$ Gorenstein ideals associated with covering maps of degree $6$. In particular one can find that such structure is given by the relations of the spinor embedding of the orthogonal Grassmann variety $\operatorname{oGr}(5,10)$.
To conclude I will talk about the work of Faenzi and Stipins for triple covers and how one can geometrically relate $r$ points on the $r$-dimensional affine space and the $r$-dimensional projective space. When applied to a triple cover $f\colon X\rightarrow Y$ it produces a birational variety to $X$ with a 'simpler' algebraic description. I will present my research on its generalisation for covers of general degree.