We study cluster categories arising from marked surfaces (with punctures and nonempty boundaries). By constructing skewed-gentle algebras, we show that there is a bijection between tagged curves and string objects. Applications include interpreting dimensions of Ext1 as intersection numbers of tagged curves and Auslander-Reiten translation as tagged rotation. An important consequence is that the cluster(-tilting) exchange graphs of such cluster categories are connected.

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We show that the scaling limit exists and is invariant under dilations and rotations. We give some tools that might be useful to show universality.

This paper provides a new construction of a supply of Darmon-like points on elliptic curves E defined over a quadratic extension of totally real number fields.

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