Email: nathan_mann@berkeley.edu
Office: CoRE Building 415
Home Institution: UC Berkeley
Mentor: Lev Borisov
I am a fourth year pure mathematics undergraduate student at UC Berkeley. As far as mathematical interests go, I am a bit of a generalist. That is to say, I have yet to take a math course that I didn't enjoy. With that being said, I have a particular affinity for algebraic geometry and mathematical logic. In my free time, I am an amateur pianist and guitarist.
Any complex surface which has the same Betti numbers as the complex projective plane but is not biholomorphic with the complex projective plane is called a fake projective plane (FPP). There are exactly 100 such surfaces, all of which can be realized as a complex 2-manifold via a quotient of the complex 2-ball B². These surfaces come in pairs, with fifty pairs in total. It is known that every FPP can also be realized as a complex algebraic surface. However, out of the fifty pairs, only about ten so far have been explicitly realized as an algebraic surface via giving an explicit set of defining polynomial equations.
The goal of this project is to find explicit defining polynomial equations for three FPPs which have not yet been explicitly realized as algebraic surfaces. In particular, these three FPPs are labeled as (C18,p=3,\{2\},D_3), (C18,p=3,\{2\},(dD)_3), and (C18,p=3,\{2\},(d^2D)_3) in the Cartwright-Steger classification. The broad outline of how we plan to do this is as follows: we start with the FPP labeled (C18,p=3,\{2I\}) in the classification, which has known defining equations. We will then attempt to lift these equations up to a common cover of these four FPPs, and then bring them back down to the three FPPs with unknown equations. This project will involve a lot of heavy computation, much of which will be carried out in Mathematica.
I would like to thank Professor Lev Borisov for mentoring me through this project. I would also like to thank the DIMACS program for providing me this outstanding opportunity to participate in this research over the summer. Lastly, I would like to thank the NSF for supporting the REU through grant CCF-2447342, and the Rutgers Math Department for providing funding for my project.
I arrived at DIMACS on Tuesday 5/26. I met with my mentor on Wednesday and Friday to discuss details of the project. We went over background information, discussed basics of FPPs, and discussed what the project would entail. Over the weekend, I began to read through chapter 10 of Artin's Algebra, to become familiar with complex representations and character theory of finite groups, as it is important to our project. In addition, I read through the paper Finding equations of the fake projective plane (C18,p=3,\{2I\}) by Lev Borisov and Bojue Wang. I also downloaded the mathematica files from the paper and began to run them on my own computer, as the computational data from these will be needed for our project this summer.
This week, I finished reading through chapter 10 of Artin. In addition, I gave a 5 minute presentation on my project on Tuesday morning to the rest of the DIMACS participants. Per Professor Borisov's GAP calculations from last week (which I doubled checked on my own computer for a quick sanity check), the automorphism group of the common cover between the three unknown FPPs and the one known FPP is C3xC3xS3, and each cover corresponds to an index 18 subgroup. Thus, the next step for us was to calculate and begin to analyze the character table for C3xC3xS3 (Artin chapter 10 is already becoming quite important). I worked with Professor Borisov on Friday to get a script set up in Mathematica to help us do this. I have also been attemtping to run the Mathematica files from the paper by Borisov and Wang, but have had some difficulty (possibly limitations in computing power), so finishing running those files is currently a big short-term goal.
I encountered some difficulty running the previously mentioned Mathematica scripts throughout this week, as they proved to be a bit finicky. After a discussion with my mentor, we decided the best thing to do would probably be to write our own scripts to run instead, as a lot of the steps from the scripts we were trying to re-use were not strictly necessary for our project. Thus, we began to work on our own scripts. The character table of C3xC3xS3 suggested there should be sections f1 and f2 in H0(18FPPHat,K) that transform via the characters (1,1,sgn) and (omega,1,sgn) respectively. We then wrote g1=f1^2, g2=f2^3, and g3=f1f2, as these products are invariant under the covering involution from the 2 to 1 cover from 18 to 9FPPHat (thus the gi descend to seection in H0(9FPPHat,2K)). The gi also satisfy g1g2=g3^2. We then wrote the gi as a linear combination of unknown coefficients over a previously calculated basis of H0(9FPPHat,2K). We subjected these to the relation g1g2=g3^2 and plugged them into previously calculated points in order to obtain a system of polynomial equations in the unknown coefficients. Our hope was to be able to solve them, and then subsequently recover f1 and f2 to give us concrete sections in H0(18FPPHat,K). Unfortunately, all of our attempts to have Mathematica numerically solve this system faield. It may be the case that we simply lack enough computing power to solve this system. For now, we are looking to see if we can find a trick to solve it with our available computing power, or for another angle to approach this problem. On an unrelated note, I took a trip to New York on Saturday the 13th, and witnessed many festivities going on in the city after the Knicks won their first NBA finals in 53 years. This was quite fun.
At the suggestion of my mentor, we decided to try a new strategy, as last week's method of trying to solve for the gi seemed like it may be beyond the capabilities of our hardware to solve. We instead began to look for the nonreduced linear cut on FPPHat. My mentor calculated the Hilbert polynomial of the known FPP mod 17 and it was as expected, thus it seemed like things may be well enough behaved over the finite field F17 to work in this space. The benefit of the finite field is as follows: there are only finitely many hyperplanes over F17, so it is possible in principle to simply do an exhaustive search to see which ones of them are nonreduced. Our only problem is that there are ~17^9 hyperplanes, and checking all of them for nonreducedness in Magma would take about a million days. To rectify this, I wrote a program in C that checked each hyperplane for a weaker but necessary condition for the hyperplane to be nonreduced. This weaker condition is still very restrictive, and also can be checked extremely quickly. The program ran in about 18 hours on my laptop, and reduced the possible candidates from ~17^9 down to ~1000. My mentor then checked the surviving 1000 hyperplanes in Magma to see if any were nonreduced, and as expected, there was exactly one such hyperplane. Our next step is to lift this hyperplane modulo 17^k, and then use the resulting 17-adic approximation to recover an explicit equation for this nonreduced cut over an algebraic number field
This week, we first finished lifting the coefficients of the nonreduced cut to an algebraic number field. I then did some more computational work with reducible cuts WiWj=0, which had been computed in a previous work. They were pairwise products of basis elements Wi and Wj of H0(9FPPHat,K). We wanted to find the connected components of these (ie, seperate WiWj=0 into Wi=0 and Wj=0). To do this, we chose a random point p on WiWj=0. Then we used Mathematica to solve WiWj=0 on the FPP in a power series near p (up to degree 30). We then looked at all quadratic monomials in our unknowns and their restriction to this power series. At this point, our original plan was to read off these series coefficients into a matrix, and then use some linear algebra from that point to find the individual equations for W1=0 and W2=0. But in the interest of saving time, we instead opted to generate a bunch of random points on W1W2=0 then use the above matrix to sort them into which lie on W1=0 and which on W2=0, which is sufficient for what we need in the next step.
This week, we construced points on the 18-fold cover of FPPHat. We started with the Wi from last week, which form a basis of H0(9FPPHat,K). We first labeled them according to their C3xC3 grading. We know that each section should have two preimages under the double covering, and we also know that if we multiply four of them together such that the gradings add up to zero, the resulting section descends to H0(FPPHat,4K). Thus, we looked for quadratic relations amongst three of the sections which we already knew, and then solved those relations together with the equations for FPPHat and an intersection with a random linear cut. By looking at solutions to these equations that did not lie on any of the three sections that we knew, it must then be the case that those solutions came from the unknown section, allowing us to get arbitrarily many points with arbitrary precision on all 17 of the Wijk. Our next goal is to extend the C2 action; for the action of C3xC3xC2 that gives the 18-fold cover, C2 is a subgroup of S3. 18FPPHat is the common cover of the four FPPs, and C3xC3xS3 is its automorphism group. FPPHat comes from quotienting by the C3xC3xC2 previously mentioned, but the three new FPPs come from quotienting by other subgroups of index 18 inside of C3xC3xS3. Thus, we need to know the full action of this group on the Wijk in order to be able to take invariants down to the three new FPPs.