In this paper we give an easy combinatorial description for the base partition ${\cal B}$ of a skew character $[{\cal A}]$, which is the intersection of all partitions $\alpha$ whose corresponding character $[\alpha]$ appears in $[{\cal A}]$. This we use to construct the cover partition ${\cal C}$ for the ordinary outer product as well as for the Schubert product of two characters and for some skew characters, here the cover partition is the union of all partitions whose corresponding character appears in the product or in the skew character. This gives us also the Durfee size for arbitrary Schubert products.