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

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... icular, 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 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 In another paper, Rana [Ran17, Theorem 1.2] assumes further that the map X → S involves blowing down exactly − 1 times and proves that, in this case, is either 2 or 3. She uses this to study the boundary of the Kollár-Shepherd-Barron-Alexeev (KSBA) moduli space for surfaces with K 2 = χ = 5 (for example, quintic surfaces). Remark 1.3. The hypothesis that p g > 0 (which is equivalent to b + (X) > 1) is there because our proof uses results from Seiberg-Witten theory and holomorphic curve theory which break down when b + (X) = 1 (or at least when the minimal model of X is rational or ruled). The hypothesis entails that the minimal model S is neither rational nor ruled but allows, for example, that S is an elliptic surface of positive genus (see Remark 1.11 below for examples).