Math260 Tutorials

1.1. Why is Algebraic Geometry Beautiful. We begin with problem number two on the homework which asks us to find 27 lines on the cubic x+y+z+1 = 0. This leads us to the following beautiful story of algebraic geometry. Every cubic in P (you can think of this as A if you want) contains exactly 27 lines. In fact, since the variety of lines in P is simple to understand (it is just the Grassmanian) we know that: • Every degree 3 polynomial in P contains 27 lines. • Every degree 5 polynomial in P contains 2875 lines. • Every degree 2d-1 polynomial in P contains a known finite number of lines. Now let’s concentrate on the degree 5 hypersurfaces in P and ask how many degree k curves are contained in these hypersurfaces. We restrict our attention to these varieties because they are Calabi-Yau which means ”very nice”. In this case the answer is much harder to figure out: • If k = 1 there are 2875 curves. • If k = 2 there are 609, 250 curves (1985). • If k = 3 there are 317, 206, 375 curves (1991). The methods used in figuring out these numbers are generalizations of the ones used for lines. However, they fail beyond very small values of k. In 1991 however, physicist in the area of mirror symmetry announced answers for all k. This lead mathematicians to try to translate their work into mathematics and lead such people as Kontsevich to develop ideas such as stable maps. Thus we see the beautiful connexions which algebraic geometry has to other areas.