General Information

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About me

I am an undergraduate at Columbia University studying mathematics. I'm primarily interested in algebraic geometry and topology, although I dabble in computer science as well.

Project Description

A fake projective plane is a complex projective surface of general type with the same Betti numbers as the complex projective plane. The first example was first constructed by Mumford as a quotient of a 2-adic analog of the complex two-dimensional ball by a finitely genrerated group. Further work by Prasad and Yeung, and Cartwright and Steger, among others, have given descriptions of all the FPPs as quotients of the complex 2-ball by a finitely generated group. There are 50 conjugate pairs of FPPs, split among 28 classes. The general theory tells us that each FPP is algebraic i.e. given by the solutions of some system of polynomial equations. However, the ball quotient description does not yield any specific equations.

My project is part of a larger program to give explicit equations of the FPPs and related surfaces of general type.

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Weekly Log

Before the REU:

I was able to meet with Professor Borisov briefly a few weeks before the program. Using Vakil, Hartshorne, and Shafaverich, I studied sheaves, invertible sheaves and line bundles, divisors, sheaf cohomology, and maps to projective space.

Week 1:

The REU officially began with move-in Tuesday and orientation Wednesday. I met with my mentor on Wednesday and learned some background on FPPs and algebraic surfaces. Topics included Kodaira dimension, the canonical class, Riemann-Roch and Serre Duality, and the Picard group. I looked at the classification of FPPs given by Cartwright and Steger and read Borisov's recent paper on a FPP. I also continued reading about surfaces from Hartshorne.

Week 2:

This week, I learned more background about surfaces of general type (with an emphasis on those with p_g=q=0) and various general results about surfaces from Hartshorne. I also read more about GIT quotients, fibrations, and ramification. Using Magma and Mathematica, I worked on computing nonreduced curves on the FPP that give torsion line bundles in order to obtain a simplification of Keum's fake projective plane.

Week 3:

I continued computing the special fibers on the FPP. The issue was that there were some poles for the defining functions on the double fibers which were difficult to figure out. THey ultimately required a clearing of suitable denominators. We also had an idea to write the 84 equations of the FPP as the Pfaffians of the minors of a 9x9 skew-symmetric matrix. A day's work of verification showed that this didn't work out, unfortunately. After computing the special fibers, we still had some difficulties with one parameter of the desired coordinate change. I decided to calculate |4H| on the FPP by computing 2nd-order neighborhoods to two special points and computing a twice-vanaishing condition on two lines on the FPP at those points.

Week 4:

Unfortunately, the calculation of |4H| wasn't useful for simplification. Fortunately, we were able to find a good coordinate change through an ad hoc method of simplifying the "simplest" coefficient of the FPP. We also decided to leverage the calculation of |4H| to find an embedding into dimension 5 projective space. I also further simplified the Keum's surface through a sort of random peturbation process on the coefficients.

Week 5:

Acknowledgements

This work was carried out while I was a participant in the 2022 DIMACS REU program at Rutgers University. I am incredibly grateful for my mentor, Lev Borisov. I am funded by NSF grant CCF-1852215.