Term for the arch supporting a gemstone

Solution 1:

Wikipedia uses the word tines:

Grafenkrone: the coronet of a Graf (count) displays nine visible tines [mounted] with pearls.

This term may not apply when the diadem has pointed arches.

The tiara below is named as a diamond spike tiara on Pinterest:

Solution 2:

The term is not specific to jewellery and is possibly more familiar as an astrological, historical or mathematical term but those points on the crown in your picture are the apexes of cusps. Merriam Webster gives one definition (sense d in the liked entry) as

an ornamental pointed projection formed by or arising from the intersection of two arcs or foils (foil here being used in the sense of "an indentation between cusps in gothic tracery")

The entry for "cusp" also has an illustration showing a piece of ornamental tracery composed mainly of cusps.

The possibly more familiar astrological and historical uses of "cusp" are derived from sense a of the MW definition

a point of transition

(in astrology the transition from one sign or house to another)

The decorative cusp is an example of a physical cusp of which there are natural and practical examples.

The definition does not preclude the case of two convex arcs intersecting as in a pointed gothic arch but the word "cusp" is more usually applied to the intersection of concave arcs as in your crown picture.