Recently I began to consider algebraic surfaces, that is, the zero set of a polynomial in 3 (or more variables). My algebraic geometry background is poor, and I'm more used to differential and Riemannian geometry. Therefore, I'm looking for the relations between the two areas. I should also mention, that I'm interested in the realm of real surfaces, i.e. subsets of $\mathbb{R}^n$.
On my desk you could find the following books: Algebraic Geometry by Hartshorne, Ideals, Varieties, and Algorithms by Cox & Little & O'Shea, Algorithms in Real Algebraic Geometry by Basu & Pollack & Roy and A SINGULAR Introduction to Commutative Algebra by Greuel & Pfister. Unfortunately, neither of them introduced notions and ideas I'm looking for.
If I get it right, please correct me if I'm wrong, locally, around non-singular points, an algebraic surface behaves very nicely, for example, it is smooth. Here's the first question: is it locally (about non-singular point) a smooth manifold? Is it a Riemannian manifold, having, for instance, the metric induced from the Euclidean space?
Further questions I have are, for example:
- Can I define geodesics (either in the sense of length minimizer or straight curves) in the non-singular areas of the surface? Can they pass singularities?
- How about curvature? Is it defined for these objects?
- Can we talk about convexity of subsets of the algebraic surface?
- What other tools and term can be imported from differential/Riemannian geometry?
I will be grateful for any hint, tip and lead in the form of either answers to my questions, or references to books/papers which can be helpful, or any other sort of help.
Best Answer
It seems to me that your interest is not in algebraic geometry, but in the differential geometry of spaces defined by algebraic equations. An algebraic variety defined over $\mathbb R$ or $\mathbb C$ is a manifold away from the singularities. The singular set is is a proper closed subset, where closed means defined by some algebraic equations, so this set is actually lower dimensional than the original object, so you have a very nice big open subset where you have a manifold. The question of extending differential geometric constructions to singularities is in general a difficult one and is the focus of a lot of research. You might be able to get a better idea by looking at complex analytic geometry. In any case, you can obviously do any of those you ask on the smooth part, but it will not be algebraic geometry. But that's OK.
One possibility you could try is indeed looking at the resolution of singularities, do your Riemannian magic there and try to get bring the results back to the original space. I suspect that you don't know what a resolution of singularities is since it is actually an very specifically algebraic geometric notion. It is the following: Let $X$ be your starting object. A resolution of singularities is a morphism $\pi:\widetilde X\to X$ such that it is an isomorphism outside a smaller dimensional subspace of $X$. You can read more about these in Lectures on resolution of singularities by János Kollár. The difficulty will be in taking whatever you do on the resolution back to the original, but the good news is that it is differential geometry, so my suggestion would be the following: For now assume that such a $\pi$ exists and see if what you want to do you can on $\widetilde X$. If so, try to see if you can "push-forward" some of those results to $X$. Perhaps you will realize that "if only $\pi$ satisfied property $P$, then I could do this" and it is possible that $\pi$ does. SO, if you get to that point, then look at Kollár's book or come back to MO and ask more specific questions.