We use bathymetry, gravimetry, and basalt composition to examine the relationship between spreading rate, spreading obliquity, and the melt supply at the ultraslow spreading Southwest Indian Ridge (SWIR). We find that at regional scales (more than 200 km), melt supply reflects variations in mantle melting that are primarily controlled by large-scale heterogeneities in mantle temperature and/or composition. Focusing on adjacent SWIR regions with contrasted obliquity, we find that the effect of obliquity on melt production is significant (about 1.5 km less melt produced for a decrease of 7 mm/a to 4 mm/a in effective spreading rates, ESR) but not enough to produce near-amagmatic spreading in the most oblique regions of the ridge, unless associated with an anomalously cold and/or depleted mantle source. Our observations lead us to support models in which mantle upwelling beneath slow and ultraslow ridges is somewhat focused and accelerated, thereby reducing the effect of spreading rate and obliquity on upper mantle cooling and melt supply. To explain why very oblique SWIR regions nonetheless have large outcrops of mantle-derived ultramafic rocks and, in many cases, no evidence for axial volcanism [Cannat et al., 2006; Dick et al., 2003], we develop a model which combines melt migration along axis to more volcanically robust areas, melt trapping in the lithospheric mantle, and melt transport in dikes that may only form where enough melt has gathered to build sufficient overpressure. These dikes would open perpendicularly to the direction of the least compressive stress and favor the formation of orthogonal ridge sections. The resulting segmentation pattern, with prominent orthogonal volcanic centers and long intervening avolcanic or nearly avolcanic ridge sections, is not specific to oblique ridge regions. It is also observed along the SWIR and the arctic Gakkel Ridge in orthogonal regions underlain by cold and/or depleted mantle.