Projective Space - Morphisms

Morphisms

Injective linear maps T ∈ L(V,W) between two vector spaces V and W over the same field k induce mappings of the corresponding projective spaces P(V) → P(W) via:

→ ,

where v is a non-zero element of V and denotes the equivalence classes of a vector under the defining identification of the respective projective spaces. Since members of the equivalence class differ by a scalar factor, and linear maps preserve scalar factors, this induced map is well-defined. (If T is not injective, it will have a null space larger than {0}; in this case the meaning of the class of T(v) is problematic if v is non-zero and in the null space. In this case one obtains a so-called rational map, see also birational geometry).

Two linear maps S and T in L(V,W) induce the same map between P(V) and P(W) if and only if they differ by a scalar multiple, that is if T=λS for some λ ≠ 0. Thus if one identifies the scalar multiples of the identity map with the underlying field, the set of k-linear morphisms from P(V) to P(W) is simply P(L(V,W)).

The automorphisms P(V) → P(V) can be described more concretely. (We deal only with automorphisms preserving the base field k). Using the notion of sheaves generated by global sections, it can be shown that any algebraic (not necessarily linear) automorphism has to be linear, i.e. coming from a (linear) automorphism of the vector space V. The latter form the group GL(V). By identifying maps which differ by a scalar, one concludes

Aut(P(V)) = Aut(V)/k∗ = GL(V)/k∗ =: PGL(V),

the quotient group of GL(V) modulo the matrices which are scalar multiples of the identity. (These matrices form the center of Aut(V).) The groups PGL are called projective linear groups. The automorphisms of the complex projective line P1(C) are called Möbius transformations.