# general questions about algebraic surfaces and Castelnuovo's contraction theorem

I am not really sure where should I ask this question so feel free to move it to other more fit community or add more tags.

My master thesis is about Algebraic surfaces and Castelnuovo's contraction theorem (it is mainly Chapter V of Hartshorne). I am asking for any question you might have that the jury might ask during the defence. Feel free to also include an answer for your question or a reference that I can look up. Examples of these questions are:

1. What are the motivations about using ample divisors? This might be seen as a natural extension of the case of curves (the projective embedding of curves is done by ample divisors)or the idea of the correspondence between globally generated divisors and morphisms to projective spaces. Do you have any other ideas or examples?

2. How much generality do we lose when we assume that we are working with smooth projective surfaces over an algebraically closed field (these are 3 assumptions)? My observation is that we always use that $$X$$ is a projective surface so that we get an ample divisor and then continue whatever we are proving. So is that it? so we could replace the projective assumption with the assumption that $$X$$ has an ample divisor? if this is true, how hard is it to find an ample divisor on general surfaces? I think Nakai-Moishezon criterion will still valid. right? What about smoothness and the algebraic closure? I notice it has been used so that we obtain a useful formula for the self-intersection number for a curve $$C$$ (that it is equal to the degree of the normal sheaf of $$C$$ on $$X$$. This, in turn, is used to prove the adjunction formula which is a central piece of the theory.

3. could you give an example of an ample divisor that is not globally generated?

• Are you asking for answers to your questions 1--3, or just more questions? The last question is answered here: math.stackexchange.com/questions/542751/… Time permitting I will say something about the others later today. Commented Jun 25, 2020 at 9:24
• @LazzaroCampeotti Thank you for pointing out these examples. I want both actually, answers to my questions (I might ass more) and more questions in general. Commented Jun 25, 2020 at 9:43

What is the general strategy for proving the classification of surfaces in characteristic zero, and what are the key technical theorems used within?

• For full disclosure, I don't know enough to give a good answer myself, but I am interested. Commented Jun 25, 2020 at 9:21

A variety is quasi-projective if and only if it has an ample line bundle. Projective means quasi-projective and proper.

There are many non-algebraic surfaces and hence no ample line bundles, some of them are known as Moisezhon surfaces.

Classification of surfaces is done for smooth projective surfaces over algebraically closed fields. There are generalizations, but is more complicated when you drop any of these hypotheses.

For example, if you want to deal with singularities, then you have to study classes of singularities, which are pretty complicated making classification unwieldy .

Algebraically closed field also is useful, since many of the classifications are done using some important numbers like genus, regularity etc. For example, smooth projective curves are classified by genus. As you probably know, a smooth projective curve of genus zero is a projective line. But over non-algebraically closed fileds, there might be two non-isomorphic ones, like projective line and the quadric in the plane.

Finally an example of a non-globally generated ample line bundle is the one associated to the divisor of a single point over a positive genus (smooth projective) curve. You can get higher dimensional examples by just taking products.