Singular algebraic curves and infinite symplectic staircases

• 成果类型: Article
• 署名作者: McDuff, Dusa; Siegel, Kyler
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01359-4
• 发表日期: 2025
• 页码: 387-459
• 关键词: embeddings; Ellipsoids; convexity; capacity; SURFACES

摘要

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 & Qopf;-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.