 This algebraic geometry lecture will be mainly about Dezague's theorem in case anyone is using subtitles Just write out how you spell that So he was a French mathematician and engineer who was one of the founders of projective geometry Anyway, his theorem is like Pappas's theorem. It's kind of remarkable that it's only about points and lines And it says the following suppose you take a point And you think at this point as being an observer and the observer is kind of observing two triangles For a couple of some lines for him to look at triangles on So here we have an observer you can think of His eye is here looking at things And he's looking at a A triangle here with three vertices a b and c and He's also drawing on some easel. So On the easel there's going to be three Points a b and c that are the images of the triangle. He sees what you do is you Join up these three triangles like this so Here I'm going to take a blue triangle and Red triangle I'm going to join up the lines bc of the Red triangle going to meet at a point Here and then I'm going to do the same thing for the a b side so side of the triangle meets this side of the capital a bc triangle and Finally, we do the same thing for the third side joining a and c so we've got an AC side going down here So we've got three points here and Pappas's theorem Says that these three points line a straight line which they don't quite in this case because I haven't drawn them very accurately So these points lie on a line So This theorem Pappas's theorem as nine lines and nine points design's theorem and Nine points design's theorem has ten lines and Ten points so this configuration of ten lines and ten points is sometimes called design's configuration and So why is design's theorem true? Well, it's a theorem about points and lines in the plane But in some sense, it's very difficult to prove if you stay inside the plane So the key point of the proof is to imagine the triangles a b c and a b c in space So we should think of this as being an Triangles in a three-dimensional space and in particular the planes containing these two triangles should be different We obviously can't do this in two dimensions because then the planes would have to be the same and now we notice That we've got a line little a little b here and the line big a big b here And we notice these lines in space must actually meet because the lines through the two a's and the lines through the two b's meet which means that these two lines are actually in the same plane so These two lines must meet at a point in space So so we would say that then the lines a b and a b meet And of course the same thing is true for the lines a c and a c and the lines b c and b c by symmetry So these points are actually well defined in space And now you notice that each of these points So the three points all lie on They all lie in the plane Containing the triangle a b c and they also line the plane containing the triangle big a b c So they lie on the intersection of these two planes which is usually a line But might not be a line because the two planes might be the same just by accident But if you choose these two planes in generic position Then their intersection will be a well-defined line. So all these three points lie on a line So if these triangles are Not both contain the same plane then desog's Theorem holds and if they are containing the same plane, then you can mumble something about taking limits and still deduce the theorem And this theorem is particularly odd because it's a theorem entirely in two dimensions and Yet you can't really in some sense you can't really prove except by moving up to three dimensions So If you've got a projective space of some dimension if it is dimension at least three it automatically satisfies desog's Theorem for the two-dimensional planes in it because you can push everything up to three dimensions If you've got a two-dimensional projective plane, it doesn't necessarily satisfy desog's theorem. So there are some examples called non-desarguin planes Which don't actually satisfy desog's theorem There's a lot of theorem turns out to be in some sense equivalent to the associativity law for multiplication. So what you can do if you've got a projective space then And you consult an old book on projective geometry They will show how you can introduce coordinates over some sort of ring for the space and desog's theorem just says That this this ring is associative and just as Papas's theorem Says that the ring is Is is commutative Ring also tends to have inverses So the result of this is that if you take a projective space which satisfies desog's theorem which is automatic and dimension at least three and Papas's theorem then it comes from Projective space over a field at least of its finite dimensional. So this shows that synthetic Projective geometry where you write down some axioms is almost the same as Analytic projective geometry where you just write down a bunch of coordinates Except for the slight problem of non-desarguin planes So an example of a non-desarguin plane might be a plane over some non-associative ring such that such as the ring of Octonians Anyway from now on we'll be forgetting about Synthetic differential geometry and just working with analytic geometry and coordinates This is a lot easier for instance if you're working in synthetic geometry and wanted to define a cubic curve You could do it, but it'd be a real headache whereas in analytic geometry. It's just trivial There's another property of differential job of Projective geometry that was noticed fairly early on which was duality for Projective space So let me give the simplest example of duality. Let's just look at a project of plane and we have these axioms any two distinct points Meet in One one line and Similarly any two distinct lines Meet in a unique point and You notice that these two axioms are dual if you swap the words point and line Now what this turns out to mean is that pretty much any theorem in The projective plane that you say about lines and points as a dual theorem about points and lines So we have a duality points Or get switched with lines So a fairly typical example is Pascal's theorem So you remember Pascal's theorem you take a conic and take six points on it and fit around with them to do something or other the dual theorem You take a conic it turns out the dual of a conic is the same as a conic and instead of taking six points on the conic you take six lines tangent to the conic and Then there's a sort of dual version of Pascal's theorem saying if you take the three Points given by intercepting these lines and pairs they all meet it. So if you take the Which way around it goes so Pascal's theorem says that if you take Three Three points given by intersecting pairs of lines They all lie on a straight line. This says if you take three lines given by Joining up pairs of points in three different ways, then they all meet at a point So One way of thinking about duality is it's just duality of vector spaces. So projective space corresponds to that the points correspond to lines of Affine space of dimension n plus one which we think of being an n plus one dimensional vector space So points of p to the n correspond to lines of this and Similarly lines in p to the n correspond to planes of 8 the n plus one and so on Planes would correspond to three dimensional subspaces and so on now for any vector space k to the n plus one You can take a dual space Of all linear transformations, which is also isomorphic to k to the n plus one although not in a canonical way and If you've got a line in a n plus one You can take its dual which would be a hyperplane just consisting of all linear transformations Vanishing on this so lines correspond to hyperplanes planes correspond to things of co-dimension to and So when you get all the way up to hyperplanes Which correspond to lines? Which are just points of projective space so duality for projective space is Very closely related just to taking the dual of a vector space