Singular algebraic curves and infinite symplectic staircases

TL;DR

This paper explains infinite staircases in ellipsoid embedding functions of del Pezzo surfaces using rational sesquicuspidal symplectic curves, introducing new algebraic curve families and methods linking algebraic and symplectic geometry.

Contribution

It introduces a novel explanation for infinite staircases via algebraic curves and develops methods for constructing singular algebraic and symplectic curves.

Findings

Infinite staircases are explained by rational sesquicuspidal symplectic curves.

New families of algebraic curves with one cusp singularity are constructed.

Methods for creating singular algebraic and symplectic curves are developed.

Abstract

We show that the infinite staircases which arise in the ellipsoid embedding functions of rigid del Pezzo surfaces (with their monotone symplectic forms) 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.