It is easy to show using birational factorization that the only curves on a surface which can be contracted to get an algebraic, smooth surface are smooth $(-1)$-curves. Furthermore, I know of examples of smooth curves of some other negative self-intersection which can be contracted in the algebraic category to result in a singular surface (where the order of the singularity is equal to the negative self-intersection?)

Question 1: Given a curve of negative self-intersection on a complex surface, what is the construction (in the analytic category) of its contraction?

 

Question 2: Given the curve, what conditions on it that determine whether the contraction is algebraic or not?

Good, clear, elementary references would be fine, as an alternative to an answer!

It is easy to show using birational factorization that the only curves on a surface which can be contracted to get an algebraic, smooth surface are smooth $(-1)$-curves. Furthermore, I know of examples of smooth curves of some other negative self-intersection which can be contracted in the algebraic category to result in a singular surface (where the order of the singularity is equal to the negative self-intersection?)

Question 1: Given a curve of negative self-intersection on a complex surface, what is the construction (in the analytic category) of its contraction?

Question 2: Given the curve, what conditions on it that determine whether the contraction is algebraic or not?

Good, clear, elementary references would be fine, as an alternative to an answer!

It is easy to show using birational factorization that the only curves on a surface which can be contracted to get an algebraic, smooth surface are smooth $(-1)$-curves. Furthermore, I know of examples of smooth curves of some other negative self-intersection which can be contracted in the algebraic category to result in a singular surface (where the order of the singularity is equal to the negative self-intersection?)

Question 1: Given a curve of negative self-intersection on a complex surface, what is the construction (in the analytic category) of its contraction?

Question 2: Given the curve, what conditions on it that determine whether the contraction is algebraic or not?

Good, clear, elementary references would be fine, as an alternative to an answer!

It is easy to show using birational factorization that the only curves on a surface which can be contracted to get an algebraic, smooth surface are smooth $(-1)$-curves. Furthermore, I know of examples of smooth curves of some other negative self-intersection which can be contracted in the algebraic category to result in a singular surface (where the order of the singularity is equal to the negative self-intersection?)

Question 1: Given a curve of negative self-intersection on a complex surface, what is the construction (in the analytic category) of its contraction?

Question 2: Given the curve, what conditions on it that determine whether the contraction is algebraic or not?

Good, clear, elementary references would be fine, as an alternative to an answer!

It is easy to show using birational factorization that the only curves on a surface which can be contracted to get an algebraic, smooth surface are smooth (-1)-spheres. Furthermore, I know of examples of smooth curves of some other negative self-intersection which can be contracted in the algebraic category to result in a singular surface (where the order of the singularity is equal to the negative self-intersection?)

Question 1: Given a curve of negative self-intersection on a complex surface, what is the construction (in the analytic category) of its contraction?

Question 2: Given the curve, what conditions on it that determine whether the contraction is algebraic or not?

Good, clear, elementary references would be fine, as an alternative to an answer!

It is easy to show using birational factorization that the only curves on a surface which can be contracted to get an algebraic, smooth surface are smooth $(-1)$-curves. Furthermore, I know of examples of smooth curves of some other negative self-intersection which can be contracted in the algebraic category to result in a singular surface (where the order of the singularity is equal to the negative self-intersection?)

Question 1: Given a curve of negative self-intersection on a complex surface, what is the construction (in the analytic category) of its contraction?

Question 2: Given the curve, what conditions on it that determine whether the contraction is algebraic or not?

Good, clear, elementary references would be fine, as an alternative to an answer!