Smooth genus-zero stacky curves with generically trivial stabilizer

A smooth genus-zero stacky curve with generically trivial stabilizer looks like P1, with finitely many analytic discs D replaced with [D/µr] for some r. (The integer r may be different at different orbifold points.) To such a curve is naturally associated a smooth integer-marked genus-zero curve, by taking the coarse moduli space and marking the image of each orbifold point P with relevant integer rP (the order of P).

Moduli
Via the map above, the moduli space M0orb is isomorphic to infinitely many copies of [M0n/G] for various n, where n is the number of orbifold points, and G is the subgroup of Sn that permutes orbifold points P1 and P2 if rP1 = rP2.

Notation
We write P1(r1,r2,...) for any smooth genus-zero stacky curve with generically trivial stabilizer with orbifold points of orders r1, r2, ....

Special cases

 * P1(2) (this is the weighted projective stack P(1,2))
 * P1(3)
 * P1(4)
 * P1(2,2)
 * P1(2,3)
 * P1(2,4)
 * P1(3,3)
 * P1(3,4)
 * P1(2,2,2)
 * P1(2,2,3)
 * P1(2,2,2,2)

Generalizations

 * Smooth genus-zero stacky curves
 * Genus-zero stacky curves with generically trivial stabilizer