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Geometry of the inversion in a finite field and partitions of ${\mathrm{PG}}(2^k-1,q)$ in normal rational curves

Let $L=\mathbb F_{q^n}$ be a finite field and let $F=\mathbb F_q$ be a subfield of $L$. Consider $L$ as a vector space over $F$ and the associated projective space that is isomorphic to ${\mathrm{PG}}(n-1,q)$. The properties of the projective mapping induced by $x\mapsto x^{-1}$ have been studied in \cite{Cs13,Fa02,Ha83,He85,Bu95}, where it is proved that the image of any line is a normal rational curve in some subspace. In this note a more detailed geometric description is achieved. Consequences are found related to mixed partitions of the projective spaces; in particular, it is proved that for any positive integer $k$, if $q\ge2^k-1$, then there are partitions of ${\mathrm{PG}}(2^k-1,q)$ in normal rational curves of degree $2^k-1$. For smaller $q$ the same construction gives partitions in $(q+1)$-tuples of independent points.