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

TL;DR

This paper constructs a specific smooth rational surface in projective 5-space with degree 11 and genus 8, highlighting its unique properties and describing the geometric configuration involved.

Contribution

It provides a new explicit construction of a rational surface with given invariants and classifies possible linear systems for such surfaces.

Findings

Construction of a smooth rational surface with degree 11 and genus 8 in P^5

Description of the blown-up point configuration in the projective plane

Identification of linear systems that characterize these surfaces

Abstract

We construct a linearly normal smooth rational surface S of degree 11 and sectional genus 8 in the projective fivespace. 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. We also present a short list, generated by the adjunction mapping, of linear systems whom are the only possibilities for other families of surfaces with the prescribed numerical invariants.