Suppose we have a quasi-projective morphism $F: V \rightarrow W$ for $V, W$ affine varieties (quasi-projective varieties isomorphic to some Zariski closed subset of affine space). I was wondering if $F$ is not only locally polynomial, but 'globally', i.e. the homogeneous polynomials that define the map on open subsets of $V$ is the same for each open subset.
I feel that this may not be the case…however, the polynomial maps defined locally have to agree on the intersections, so there is some restriction to defining any such map.
How do I show that from the definition of a quasi-projective morphism that such a morphism between affine varieties is described as a 'global' polynomial?
Best Answer
No, this is impossible: any map of affine varieties $X\to Y$ is equivalent to a map of their coordinate algebras $k[Y]\to k[X]$, which provides you the "global" polynomial description you seek to avoid.