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July 17, 2023
My girlfriend recommended the band Zelda to me two days ago. She acted astonished when I said I’d never listened to them before, which is what she does every time I don’t know about something she brings up. I listened to three whole albums of Zelda while I was doing problems from Shafarevich’s Basic Algebraic Geometry at the library after my class was over. Shafarevich is very much an elementary book, and I feel a little embarrassed that I have to resort to reading it, but it has lots of very concrete problems on subjects like divisors and differential forms that I’ve learned at a very abstract level from Hartshorne, the standard introduction to modern Algebraic Geometry, but don’t really have the intuition for how to work with.
The class I’ve been taking at Fudan is a two week summer course that covers three different areas of current research. Last week was entirely about K-Stability. The author of the linked book, Chenyang Xu, led the lectures. He’s one of the leading researchers in the field. Other than the 4 or so different interlocking ways to define K-Stability, I didn’t really understand anything from his course. Instead I more or less compiled a long list of things I need to learn or be more fluent at if I ever wanted to learn K-stability seriously (which might not actually be the case — I knew nothing about it before the course, and may very well return to never thinking about it again).
One thing I noticed was how many of the ways divisors were used in various definitions and proofs were things it had never occurred to me divisors could be used for. There’s a correspondence between equivalence classes of divisors and line bundles, and I’ve by this point gathered line bundles are omnipresent and of extreme importance, so of course it would make sense that divisors are important too.
As a result one of the main takeaways of the first week of the course is that I need to learn how to actually use divisors, instead of just knowing the definitions and basic theorems. So I figured I might as well start from the beginning, and do all the exercises in the three sections of Shafarevich devoted to divisors (one on the general theory, one of divisors of curves, and one specifically about elliptic curves). Then I figure I’ll do a rereading of Hartshorne’s chapter on divisors, which might take on new meaning now that I know a lot more about how they fit into the broader world of Algebraic Geometry.
For those who have no idea what I’m talking about, a divisor is basically a way of representing the intersection of varieties. A variety is the main thing that Algebraic Geometry is concerned with. It’s essentially the “shape” defined by a (multi-variable) polynomial. So for example, a single equation with two variables defines a curve, which is one dimensional. The most basic examples you might remember from high school are the line (y=x) and the parabola (y=x^2). With three variables you define a surface, which is two dimensional. Two curves intersect at a collection of points (zero-dimensional objects) and two surfaces intersect at a collection of curves (one-dimensional objects). The main thing to notice is that the dimension goes down by one with each intersection.
There is also the concept of multiplicity. If you think back to the parabola y = x^2, a horizontal line above the x-axis always intersects with two points. As the line moves down those two intersections move closer and closer together until becoming a single point (0,0) at the x-axis. There is something special happening at origin, related to the square in the equation y = x^2, and we say that point has multiplicity 2, rather than multiplicity 1 like the other points of intersection had. Multiplicities and intersection numbers are a somewhat subtle topic, so I’ll just leave the explanation at that without going into any more details.
So in order to represent intersections, we need the shapes of the intersection, which are sub-varieties, and the multiplicities of each intersection. If you are from the computer science world, you can think of a divisor as a datatype that is just a list of sub-varieties and multiplicities. We can add two divisors together by adding the multiplicities for each sub-variety (the sub-varieties are called prime divisors), and in this way we form an abelian group. Perhaps you’ve heard about elliptic curves being important for number theory (the first time I ever heard about an elliptic curve was in high school when reading an extremely simplified explanation of Wiles proof of Fermat’s Last Theorem). It’s ultimately this abelian group structure that allows for this connection.
Anyway, divisors give all sorts of information about a variety. Intersections give a sort of local picture, like a 2D cross section of a complicated 3D shape that can’t be visualized all at once. Divisors are important in the classification of varieties. The difference between two birational varieties is manifested in divisors. Divisors can also be used to define a measurement of how bad a singularity (a point where the variety is badly behaved) is.
Listening to Zelda, I enjoyed the progression from rock to weird genreless music, something between children’s song and choral at the end of the world. I’m reminded of the likes of Hirono Mio’s EP Fruits of the Moon or Tanoshii Ongaku. I suppose Zelda is somewhere in the middle, with much more rock DNA, even though it’s not immediately apparent by their third album.
Instead of having lunch today at Fudan, I took the subway to Wujiaochang and got curry there. I like the midday subway feeling. Normally I only take the subway in the mornings or the evenings. However, if you take it around noon there’s less people. It feels like I’m on vacation — a weekday vacation while everyone else is working. I miss the days when I had nowhere to be, and instead was terrified about my future because I had no job. Often that terror would be hidden deep enough inside of me that I could walk around the city, take the subway, explore new places, and forget the precariousness of my situation. Nowadays whenever I have time to do whatever I want, I can’t forget how limited that time is, how my moments of freedom are rapidly dripping away as appoints and places I need to be rapidly approach. I don’t even have that demanding of a schedule, but I still feel this way constantly. I’m not sure what to do about it.