Unwanted shadow effect on points

When using a points texture, if two points overlap there is a line of a different color drawn between them. This line isn’t in the texture, which is a white circle (ffffffff) with transparent black background (00000000).
It’s visible in this image as a dark arc between the two lighter colored points.
What’s causing the line, and is there a way to avoid it?

How do you configure your points material?

var material = new THREE.PointsMaterial({size,
map: sprite,
transparent: false,
alphaTest: 0.5,
vertexColors: true});

Taken from some example. I discovered as a workaround that if I use alphaMap instead of map, the line goes away. I still don’t understand what’s creating the line when I use map.

Does it disappear if you increase the value of alphaTest? Try it with 0.75 or even higher.