I am reading Silverman's Advanced topics in the arithmetic of elliptic curves. In the context of schemes, a point is defined to be regular or non-singular if its tangent space has dimension equal to the Krull dimension of its local ring. Then there's this example, which implies the existence of singular points which are regular:
Are singular points not supposed to be non-regular by definition? This seems contradictory to me.
[EDIT] I suppose the use of 'singular' here refers to the special fibre but not to the whole scheme, which could be non-singular, i.e: a model for the curve.