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Linear orbits of $$d$$-tuples of points in $$\mathbb{P}^ 1$$. (English) Zbl 0781.14036
Summary: This paper is concerned with the orbits of the natural action of PGL(2), the automorphism group of the projective line, on the projective space of dimension $$d$$ parametrizing $$d$$-tuples of points on the line. The closure of each orbit is a projective subvariety of which the degree, the boundary and the multiplicity at points of the boundary are determined. This is used to classify the smooth orbit closures as well as the orbit closures smooth in codimension one.

##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 14M17 Homogeneous spaces and generalizations
##### Keywords:
orbits of action of PGL(2); orbit closures
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