Sammendrag
This thesis is a study of linearly normal smooth rational surfaces of degree eleven and sectional genus eight in the projective fivespace. Surfaces satisfying these numerical invariants are special, in the sense that the first cohomology group of the sheaf O(1) on such surfaces has positive dimension.
Our main result is a proof of the existence of a family of surfaces satisfying the prescribed numerical invariants above. The construction is done via linear systems and we describe the configuration of points blown up in the projective plane to obtain this rational surface. A corollary of this result is that there exists a smooth rational surface with Hilbert polynomial $P(n)=11\binom{n}{2}+4n+1$.
We also present a list, generated by the adjunction mapping, of linear systems whom are the only possibilities for other families of surfaces with the prescribed numerical invariants.
A continuation of this thesis has later been published as the article “Smooth Rational Surfaces of d=11 and pi=8 in P5” in Math. Scand. 119 (2016), no. 2, pp 169-196.