We study endomorphism algebras of 2-term silting complexes in derived categories of hereditary finite dimensional algebras, or more generally of Ext-finite hereditary abelian categories. Module categories of such endomorphism algebras are known to occur as hearts of certain bounded t-structures in such derived categories. We show that the algebras occurring are exactly the algebras of small homological dimension, which are algebras characterized by the property that each indecomposable module either has injective dimension at most one, or it has projective dimension at most one.
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.