Smooth Rational Surfaces Of $d=11$ And $\pi=8$ In $\mathbb{P}^5$

Abstract

We construct a linearly normal smooth rational surface $S$ of degree $11$ and sectional genus $8$ in the projective five space. Surfaces satisfying these numerical 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 configuration of points blown up in the projective plane. Using the theory of adjunction mappings, we present a short list of linear systems which are the only possibilities for other families of surfaces with the prescribed numerical invariants.