Projection of five lines in projective space

Abstract

The moduli space of five lines in $\mathbf{P}^2$ can be described by a quintic Del Pezzo surface in $\mathbf{P}^5$. Given five fixed lines in $\mathbf{P}^3$ and a fixed plane, we define a map from $\mathbf{P}^3$ to the quintic Del Pezzo surface by projecting the lines to the fixed plane, and taking the point on the Del Pezzo surface defined by the image lines as the image of the point of projection. We show that the fibers of this map are twisted cubic curves. Conversely, we show that the moduli space of curves in $\mathbf{P}^3$ with the five fixed lines as secants can be seen as isomorphic to the quintic Del Pezzo surface.