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We construct a linearly normal smooth rational surface S of degree 11 andsectional genus 8 in the projective fivespace. Surfaces satisfying thesenumerical invariants are special, in the sense that $h^1(\mathscr{O}_S(1))>0$.Our construction is done via linear systems and we describe the configurationof points blown up in the projective plane. We also present a short list,generated by the adjunction mapping, of linear systems whom are the onlypossibilities for other families of surfaces with the prescribed numericalinvariants.

Smooth rational surfaces of $d=11$ and $\pi=8$ in $\mathbb{P}^5$