One of my hobbies is collecting examples of theorems which are geometry in the sense that Euclid would understand the term, yet illustrate ideas of modern algebraic geometry. I’m going to start posting explanations of some of these results here. My first example is a nineteenth century result known as Poncelet’s Porism.
Let C and D be two conics. Carry out the following process. Take a point on C. Draw a tangent to D through , and let be the other point where this tangent meets C. Now, draw the other tangent to D passing through and let be the point, other than , where this tangent line meets C. Continuing in this fashion, define , and so forth. Now, suppose that . In that case, if we had started the process at some other point on C, and defined , , in the same manner, then we would also have .
Here is an excellent animated picture, from Mathworld, of the points sliding around as we move .
From a modern perspective, Poncelet’s Porism is interesting because it is one of the few theorems of classical geometry which involves a non-rational variety — namely a genus one curve.
One of the basic tools of modern algebraic geometry is to set up incidence varieties which describe how various figures relate to each other. In this case, let X be the space of pairs so that the point p is contained in the line , the point p is contained in C and the line is tangent to D. (One of the great things about the modern approach is that I can talk about this object without discussing what projective space it lives in.) Let be the involution of X which sends to where is the other point, besides p, where meets C. Similarly, let be the involution sending to where is the other tangent, besides , to D passing through p. Our geometric construction is just repeatedly alternating between and . What we want to show is that, if has a fixed point, then must be the identity.
In differential geometry, X is simply a disjoint union of two circles (if the conics are nested), and there is no reason that might not have some complicated collection of fixed points. However, in algebraic geometry, we get to see these two circles as the real points of a complex curve, and the complex geometry of this curve is much more restrictive. So, let us now think about the construction of X with complex coordinates instead of real ones and ask what kind of curve X is. Let denote the dual conic to D, that is, the space of lines tangent to D. Then, by definition, X embeds into . Recall that a conic is a genus zero curve, which we might as well think of as , so X embeds into .
Next, check that X is smooth. This is trickier than it seems, because once we move to looking at complex points, there are four points of X where projection onto C does not give a local coordinate. Namely, at the preimages of the four intersection points of C and D. Similarly, there are four points at which projection onto does not give a local coordinate. (Can you see what they are? Hint: draw the two conics to meet at four real points and the picture will be easier to visualize.) When the conics are drawn in a nested configuration, these eight bad points on X have complex coordinates, so it is easy to ignore them, but they do affect the geometry. In any case, it turns out that, at every point of X, either projection to C or projection to gives a local coordinate, so X is smooth.
Now, over a generic point of C, there are two points of X. And, over a generic point of , there are also two points of X. So X is a smooth hypersurface in of degree (2,2). Here’s an exercise for those who’ve just taken an algebraic geometry course: check that X is a curve of genus one. Here is a harder exercise: obtain the same result by embedding X into the flag variety of pairs .
Now, a genus one curve is topologically a torus . If we pick a point on X to call the origin, then it naturally acquires an abelian group law, where the group is also . In terms of this group law, every involution of X is of the form for some . So, for some y and for some z. We compute that . In other words, is a translation on X. So is also a translation on this group. In any group G, if translation by g has a fixed point, then g is the identity element and every point is fixed under translation by G. This was the desired conclusion. QED
Two final notes. First, choosing a base point to call zero on X was unnecessary. The morally correct argument proceeds in terms of principal homogenous spaces. Exercise: give this argument.
Second, there is a very interesting classical problem posed by Poncelet’s porism when the two conics are circles. Let the radii of the circles be R and r, and the distance between their centers be d. Give conditions, in terms of R, r and d, for when the construction cycles with period N. One of the nifty things that we get almost for free from modern algebraic geometry is that this relation will be given by a polynmial. For N=3, the explicit polynomial is ; this formula was found by Euler and is proven here. The formulas grow more and more complex as N grows; Mathworld has a list for N up to 20, with some gaps. I notice that the formula for N=11 was much harder to find than those for 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18 and 20. The modular curve , which parameterizes genus one curves with a translation by a point of order 11, is not rational, whereas the modular curves for all of these other values of N are. (I am cheating a bit in my description of .) This is surely not a coincidence…