Let X be a Noetherian separated scheme. Let E ⊂ X be an effective Cartier divisor such that there is an isomorphism E → P1k where k is a field. Then we say E is an exceptional curve of the first kind if the normal sheaf of E in X has degree -1 on E over k.
You can get an example of the situation above by starting with a Noetherian separated scheme Y and a closed point y such that the local ring of Y at y is a regular local ring of dimension 2 and taking the blowup b : X → Y of y and taking E to be the exceptional divisor.
Conversely, if E ⊂ X is gotten in this manner we say that E can be contracted.
The following questions have been bugging me for a while now.
Question 1: Given an exceptional curve E of the first kind on a separated Noetherian scheme X is there a contraction of E?
Question 2: Given an exceptional curve E of the first kind on a separated Noetherian scheme X is there a contraction of E but where we allow Y to be an algebraic space?
Question 3: Suppose that Y is a separated Noetherian algebraic space and that y is a closed point of Y such that the henselian local ring of Y at y is regular of dimension 2. Is there an open neighbourhood of y which is a scheme?
Question 4: With assumptions as in Question 3 assume moreover that the blow up of Y in y is a scheme. Then is Y a scheme?
In these questions the answer is positive if we assume that X or Y is of finite type over an excellent affine Noetherian scheme (and I think in the literature somewhere; I’d be thankful for references).
But… it might be interesting and fun to try and find counter examples for the general statements. Let me know if you have one!
Just so I understand, is the ambient scheme regular of dimension 2, and does it contain $E$ as a Cartier divisor?
Sorry, I forgot to say that E is an effective Cartier divisor; I have edited the post. The fact that E is an effective Cartier divisor implies that every local ring of X at a closed point of E is regular of dimension 2. But part of the problem with not assuming more about X is that it need not be the case that X has dimension 2 in an open neighbourhood of E, although I have no counter example.