Bounds on Wahl singularities from symplectic topology

Abstract

Let X be a minimal surface of general type with p g > 0 (equivalently, b + > 1) and let K 2 be the square of its canonical class. Building on work of Khodorovskiy and Rana, we prove that if X develops a Wahl singularity of length in a Q-Gorenstein degeneration, then 4K 2 + 7. This improves on the current best-known upper bound due to Lee 400 K 2 4 . Our bound follows from a stronger theorem constraining symplectic embeddings of certain rational homology balls in surfaces of general type. In particular, we show that if the rational homology ball B p,1 embeds symplectically in a quintic surface, then p 12, partially answering the symplectic version of a question of Kronheimer.is a continued fraction expansion. The number is called the length of the singularity. The index of the singularity is p and is bounded above by 2 .Theorem 1.1. Let X be a minimal surface of general type with positive geometric genus p g > 0 which has a finite set of Du Val and Wahl singularities. Then the length of any of the Wahl singularities satisfies 4K 2 X + 7 .Remark 1.2. The best bound for Wahl singularities we could find in the algebraic geometry literature 1 is due to Lee [Lee99, Theorem 23], who showed that if X is a stable surface of general type having a Wahl singularity of length , and if the minimal model S of the minimal resolution X of X has general type, then 400 K 2 X 4 .

Optimal bounds for T-singularities in stable surfaces

“…On the other hand, bounds for each singularity in W would be weaker than bounds which consider all singularities at once. For example, by the results of Evans and Smith [ES17], we have that r i ≤ 4K 2 W + 1 when all d i = 1 and W is not rational. (This is essentially the bound in [RU17] when we have one Wahl singularity.)…”

Geography of minimal surfaces of general type with Z22$\mathbb {Z}_2^2$‐actions and the locus of Gorenstein stable surfaces

“…Nevertheless, it may happen that it is also closed in 𝔐 𝐾 2 ,𝜒 , that is, it may happen that 𝔐 𝐺𝑜𝑟 𝐾 2 ,𝜒 is a union of connected components of 𝔐 𝐾 2 ,𝜒 . Work of Anthes, Evans, Franciosi, Pardini, Rana, Reyes, Rollenske, Smith, Urzúa, and others suggests that this is never the case (e.g., [3,9,11,23,24]). In this note, we are able to prove the following:…”

“…Nevertheless, it may happen that it is also closed in $$\overline{\mathfrak {M}}_{K^2, \chi }$$, that is, it may happen that $$\overline{\mathfrak {M}}^{Gor}_{K^2, \chi }$$ is a union of connected components of $$\overline{\mathfrak {M}}_{K^2, \chi }$$. Work of Anthes, Evans, Franciosi, Pardini, Rana, Reyes, Rollenske, Smith, Urzúa, and others suggests that this is never the case (e.g., [3, 9, 11, 23, 24]). In this note, we are able to prove the following: Theorem Let $$(K^2, \chi )$$ be an admissible pair such that $$2\chi -6\le K^2\le 8\chi -8$$.…”