Darboux transformations of spacelike curves in the Lorentz-Minkowski plane

This paper concerns (Lorentz-)Darboux transformations in the Lorentz-Minkowski plane $\mathbb{R}^{1,1}$. We use the Penrose diagram for conformal compactification and show some unique properties of Darboux transformations of spacelike curves in $\mathbb{R}^{1,1}$, especially with regard to singularities and blowup. Finally, we make some comments on Darboux transformation of type-changing curves.