Ifd is a positive integer thenP(a0,a1,...,an) is isomorphic toP(da0,da1,...,dan). This is a property of theProj construction; geometrically it corresponds to thed-tupleVeronese embedding. So without loss of generality one may assume that the degreesai have no common factor.
Suppose thata0,a1,...,an have no common factor, and thatd is a common factor of all theai withi≠j, thenP(a0,a1,...,an) is isomorphic toP(a0/d,...,aj-1/d,aj,aj+1/d,...,an/d) (note thatd is coprime toaj; otherwise the isomorphism does not hold). So one may further assume that any set ofn variablesai have no common factor. In this case the weighted projective space is calledwell-formed.
The only singularities of weighted projective space are cyclic quotient singularities.
The weighted projective spaceP(a0,a1,...,an) is isomorphic to the quotient of projective space by the group that is the product of the groups ofroots of unity of ordersa0,a1,...,an acting diagonally.[2]
^M. Rossi and L. Terracini, Linear algebra and toric data of weighted projective spaces. Rend. Semin. Mat. Univ. Politec. Torino 70 (2012), no. 4, 469--495, proposition 8
