So, I WAS going to give a classical proof of a classical algebraic geometry theorem: that there are exactly twenty-seven lines on any smooth cubic surface. I wanted to avoid using the machinery of divisors and all the attached technicality, but the proof I came up with was rather nasty, needs a lot of lemmas, almost all of which are technical. So instead, I’m going to talk about my understanding of the development of the subject.
I’m going to assume that virtually nothing is known by the reader, so that even the less mathematical readers can follow along.
Like most mathematical histories, we should start looking to ancient Greece. The Greeks looked seriously at conic sections, and were, in particular, fascinated by ellipses. However, the Greeks did know the full real classification of conic sections from about 200 BC, a point, a pair of lines intersecting, a parabola, a hyperbola, and an ellipse.
The next really big advancement came with Descartes, who brought us what is now called the Cartesian coordinate system. The simple idea of assigning to each point in the plane two numbers turned out to be extraordinarily powerful. With it, we gained a fairly straightforward way of manipulating -dimensional space, as the points could be thought of as just -tuples of numbers. In particular, it was now possible to give algebraic characterizations of many geometric objects. For instance, conic sections turned out to be the solutions to polynomials of the form , with the coefficients determining which one you got (or if you got the empty set by accident, which can also be thought of as a conic).
Simultaneously with Descartes, the notion of complex number began to arise from algebra. The first significant appearance is in the case of quadratic polynomials which can be solved by a well-known formula. Sometimes, the number under the root is negative, and this hints that complex numbers should come in. However, mathematicians of the time were still somewhat leery of even the negative numbers, and these quadratics had no REAL solutions, so it was just taken as a sign of nonexistence.
Alas, the cubic polynomial was a harder nut to crack. Eventually, however, Tartaglia was able to work out a formula that worked for many cubics. However, when he applied it to solving the simple cubic , he suddenly had square roots of negative numbers! And yet, this polynomial has three distinct real solutions. Worse, when Cardano managed to solve the general cubic, and had a student who solved the quartic, and both solutions required the extensive use of complex numbers. This brought humanity into contact with its first algebraically closed field though this had not been discovered completely yet.
Yet again at the same time, Desargues developed the notion of projective space. The idea here is that, in art, when you have two parallel lines in perspective, it looks like they are going to intersect, but not until they are infinitely far away. Also, in Euclidean geometry, any two lines intersect…unless, of course, they run parallel to each other. Exceptions are abhorrent in mathematics, because they make results far less elegant. They are something to be tolerated if they cannot be fixed. Well, projective space fixed this, or more precisely, the projective plane does. The intuitive notion is just that you add a point in each direction where lines parallel to each other intersect. To be rigorous, you call the projective plane the set of all lines in three-dimensional space that pass through the origin. This works out because if you look identify all the lines with their intersection with a plane one unit above the -plane, you get another copy of the plane, but there are lines left over, and they become the points at infinity. This construction works out for higher dimensions too, which is one of the reasons it’s a good way to think about it.
If you think that all of this indicates that the late 16th and early 17th centuries were exciting for algebraic geometry, you’d be quite correct. This period of time essentially started the subject, because algebra and geometry had never before been more closely woven together. For one thing, looking in the projective plane, it turns out that the extra points you get at infinity make the ellipse, the hyperbola, and the parabola all into the same conic! Though this still leaves the empty set, the single point, a pair of lines, and even just a single line as degenerate versions of the quadratic polynomial in two variables. Combining the notion of complex numbers with that of projective space, however, can solve this. Instead of looking at , the set of -tuples of real numbers and looking at real lines through the origin, we look at and look at complex lines. This gives us complex projective space. In complex projective space, the empty conic and the point become examples of the others, and we can break down conics into precisely three types: the smooth conic, a pair of lines, and a double line, and this is the full classification that we use today.
The impression you should come away with from the example of conics is that complex projective space makes things a lot nicer. As a quick example of this, let’s look at the complex projective plane. Now, one of the difficulties of projective space is that it increases the number of variables that need to be kept track of, because a function on , the -dimensional complex projective space, is the same as a homogeneous function on . All homogeneous really means, is that the function is constant on lines through the origin. So now our plane curves are given by polynomials in three variables, , and the homogeneity condition for polynomials says that all the terms are of the same constant degree, which is called .
Now, we take two plane curves, and . First thing first, we require that and are irreducible polynomials, that is, they cannot be factored (really, all we must do is require that the have no factors in common). What this does is make sure that there is no curve in the plane on which they both vanish. So now we look at the intersection of the curves and . The intersection is given by the common solutions to both polynomials, and a fairly simple thing in algebraic geometry is to show that two plane curves intersect in a finite number of points. The question remaining is: how many?
Intersection theory is rather important, but I don’t have time to go into the details here, so I’ll just say that there is a notion of intersection number such that it will tell you when two curves intersect and you should count the point of intersection more than once, and which, when the curves intersect transversely, that is, their tangent vectors at the point span the plane, the intersection number is one. So a theorem of Bezout‘s, states that the number of points (with multiplicity taken into account via the intersection number) in the intersection of these two curves will always be . This theorem is nifty in other ways, and to see one of them check out this post at the Everything Seminar.
This post has gotten rather long already, so I think I’ll stop there, and pick up where I left off some future week by describing the development in the late 19th century and into the 20th century.