  We show that the infinite staircases which arise in the ellipsoid embedding
functions of rigid del Pezzo surfaces can be entirely explained in terms of
rational sesquicuspidal symplectic curves. Moreover, we show that these curves
can all be realized algebraically, giving various new families of algebraic
curves with one cusp singularity. Our main techniques are (i) a generalized
Orevkov twist, and (ii) the interplay between algebraic $\Q$-Gorenstein
smoothings and symplectic almost toric fibrations. Along the way we develop
various methods for constructing singular algebraic (and hence symplectic)
curves which may be of independent interest.
