How to describe morphisms to a weighted projective space (bundle)?

Question

The case of an usual projective space (bundle) is well known (Grothendiek,EGA II, Publ.Math. IHES, 8, 1961; or Hartshorne, Alg.Geom.).The more general case of toric varieties has been considered by D. Cox (for smmooth vatieties),Tohuku Math. J. 47(2)1995 , and by T.Kajiwara (non smooth case), Tohuku Math. J. 50(1)1998. I am interesed in characterizing morphisms to weighted projective spaces (bundles) , à la Grothendieck. No reference is known to me . Remark : linked to my question is "MO question/answer: Maps to projective space == line bundles; what do maps to weighted projective space correspond to?" (July 12, 2014).

Answer 1

Let S be an A-polynomial graded algebra, generated by n+1 variables of degrees grater or equal to 1 . Denote u the structural morphism X=Proj(S)->Y=Spec(A).We have a morphism u*(S)->⊕F(i) (i≽0), where F is the structural sheaf of X. This is an epimorphism if S is generated in degree 1.So, first we recall the usual case: 1) Case where S is generated in degree 1 . Proj(S) satisfies the following universal problem.a)A morphism over Y: g:Z->X (whith f:Z->Y) induces an epimorphism f*(S)->⊕(L⊗...⊗L), where ⊕ is over i≿0, ⊗ is taken i times and L=g*(L)(invertible sheaf over Z).b)Conversely,assume given a graded epimorphism: f*(S)->⊕(L⊗...⊗L)=:W, where L is an invertible sheaf over a Y-scheme Z (via f:Z->Y). This induces a Z-morphism : Z=Proj(W)->Proj(f*(S))=ZxX. Project and get a morphism g:Z->X (over Y). 2)Weighted case : Notation is similar to 1). Let m be the lcm of the weights (i.e. degrees of the variables) and s their sum. Fix any integer r, multiple of m, and greater than nm-s=:t. There is epimorphism u*(S^(r)):->⊕F(jr)(j≿0), where F(jr)=⊗F(r)(⊗taken j times), and F(r) is invertible since m divides r.a) Given a morphism g:Z->X (over Y as in 1)), we obtain an epimorphism f*(S^(r))->⊕(L⊗...⊗L) (notaion as above),with L=g*(F(r))(invertible over Z).b)Conversely,an epimorphism f*(S^(r))->⊕(L⊗...⊗L) where L is an invertible sheaf over a Y-scheme Z (via f), induces a morphism of Y-schemes Z->X (in the same way as 1)b), since Proj(S^(r))=Proj(S)).