The processes that lead to the fourfold variation in arc-averaged compositions of mafic arc iavas remain controversial. Control by the mantle-wedge thermal structure is supported by chemical correlations with the thickness of the underlying arc crust(1-3), which affects the thermal state of the wedge. Control by down-going slab temperature is supported by correlations with the slab thermal parameter(3-7). The Chilean Southern Volcanic Zone provides a test of these hypotheses. Here we use chemical data to demonstrate that the Southern Volcanic Zone and global arc averages define the same chemical trends, both among elements and between elements and crustal thickness. But in contrast to the global arc system, the Southern Volcanic Zone is built on crust of variable thickness with a constant slab thermal parameter. This natural experiment, along with a set of numerical simulations, shows that global arc compositional variability is dominated by different extents of melting that are controlled by the thermal structure of the mantle wedge. Slab temperatures play a subordinate role. Variations in the subducting slab's fluid flux and sediment compositions, as well as mantle-wedge heterogeneities, produce second-order effects that are manifested as distinctive trace element and isotopic signatures; these can be more clearly elucidated once the importance of wedge thermal structure is recognized.