A Criterion for Closed Immersions and Applications

Example 1.1. Let C ⊆ Ak be the nodal cubic plane curve given by y 2 = x + x, where k is some algebraically closed field. Let C̃ be its normalization; then C̃ ∼= A. Finally, let D be the open subcurve of C̃ obtained by removing one of the two points lying over the node of C. We consider the morphism D → Ak. Via an explicit parametric form, it is easy to check that this morphism is injective both on points and on tangent spaces. However, we also see that the map is not an immersion: for instance, an immersion would give an isomorphism of D onto its image, but while D is non-singular, its image has a node.