Mattie Ji

General Information

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Name: Mattie Ji
Email: mattie_ji[at]brown[dot]edu
REU Advisor: Lev Borisov
Home Institution: Brown University
Project: Simplifying explicit equations of fake projective planes
Link to my CV
Link to my xxx archive papers
Link to my GitHub

Project Description

A fake projective plane $X$ is a smooth complex surface with the same Hodge numbers as $\mathbb{C}P^2$ but not biholomorphic to it. The Kodaira dimension of any fake projective plane is $2$ and hence $X$ is an algebraic surface of general type. It follows as a consequence of the Calabi-Yau Theorem that $X$ may be realized as the quotient of $\mathbb{B}^2/\pi_1(X)$ where $$\mathbb{B}^2 = \{(z_1, z_2) \in \mathbb{C}\ |\ |z_1|^2 + |z_2|^2 < 1\}$$ is the complex unit $2$-ball.

By the works of Cartwright and Steger and Prasad and Yeung, it is known that there are exactly $100$ fake projective planes by analyzing the quotients above. However, for many of these surfaces, explicit embedding described by algebraic equations are unknown. For the ones that do have explicit equations written, the equations tend to be very complicated.

The goal of my REU is to simplify these explicit equations and (as plans often change) find explicit equations for new pairs of fake projective planes. See my initial presentation for a more detailed introduction.

Research Log

Week 1: 5/30-6/4

My first in-person meeting with Professor Borisov was on Thursday (6/1). This week, I focused on learning and reviewing some background material involved in this project. Topics included the Riemann-Roch Theorem, Noether's Formula, the Picard Group, Serre Duality, and the Holomorphic Lefschetz fixed-point formula.

I also read the past works done by a former REU student who worked with Professor Borisov - Zachary Lihn - in their joint paper Realizing a Fake Projective Plane as a Degree 25 Surface in $\mathbb{P}^5$. There's a lot in the paper and its accompanied code that I was unfamiliar with, so I decided to start re-implementing what Zachary Lihn and Professor Borisov did step by step myself.

On a side note, during this time, I found a very cool proof of Riemann-Hurwitz formula using Euler calculus in this paper by Gusein-Zade.

Week 2: 6/5-6/11

On Monday, I delivered a short 5 minute presentation describing my project. After that, we decided that we will focus on embedding the fake projective plane $X = (a = 7, p = 2, \emptyset, D_3 2_7)$ in the Cartwright-Steger classification to $\mathbb{C}P^5$. Previously, this surface has been implemented into $\mathbb{C}P^9$ by Borisov and Keum in their work Explicit equations of a fake projective plane.

I started working with methods to implement this. The overall plan we came up is as follows - there exists an ample divisor $H$ on $X$ such that its canonical divisor $K$ is linearly equivalent to $3H$. By a combination of the Riemann-Roch Theorem and the Kodaira Vanishing Theorem, we can find that, for $n \geq 4$, \[\dim_{\mathbb{C}} H^0(X, nH) = \frac{(n-1)(n-2)}{2}\] Previously, Borisov and Keum were able to embed $X$ into $\mathbb{C}P^9$ by constructing the $10 = \dim_{\mathbb{C}} H^0(X, 6H)$ global sections for $6H$. Our goal is to construct the basis of $6$ global sections for $5H$ to construct a map into $\mathbb{C}P^5$ and verify that it is an embedding.

The torsion component of $Pic(X)$ is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^4 = (\mathbb{Z}/2\mathbb{Z}) D \oplus (\mathbb{Z}/2\mathbb{Z})^3 D_1$, where the latter summand is generated by a $C7$ action on $D_1$ as runs through all non-trivial elements in the summand.

Since the original embedding used $6H$, this means that sections of $12H$ are homogeneous quadratic terms in terms of the variables of the embedding for $6H$. Hence, one idea is to follow what Zachary Lihn and Professor Borisov did in the previous REU and to consider $12H = (5H + D) + 4H + (3H + D)$. As long as we can generate random points vanishing on $4H$ and $3H + D$, we can find quadratic polynomials vanishing on both $4H$ and $3H + D$. The basis of that space should be dimension $\dim_{\mathbb{C}} H^0(X, 5H + D) = \frac{3 \cdot 4}{2} = 6$, and this will be our desired sections.

I worked on finding equations representing $4H$ first. The idea for this part is the same as Zachary Lihn and Professor Borisov's work last year. By the Holomorphic Lefschetz Formula, we can find that \[H^0(X, 4H) \cong \mathbb{C} r_3 \oplus \mathbb{C} r_6 \oplus \mathbb{C} r_5\] where $r_3 \to r_6 \to r_5 \to r_3$ is permuted by a $C3$ action. We can actually realize $r_3^3$ and $r_3 r_5 r_6$ as quadratics in $12H$. Furthermore, we can compute relations on $r_3^3$ and $r_3 r_5 r_6$ by finding the order two neighborhoods of them around points that we know vanish on $r_3$.

It turns out that a student who has worked with Professor Borisov in the past - Yanxin Lin - have already computed the equations of 3H + Torsion in $6H$. (The equations in $6H$ are technically the doubled sections of the section tensoring with itself, but in programmatic calculations, random points found on the curve will be on 3H + Torsion.) So we asked him to join the project as well.

Week 3: 6/12-6/18

On Monday, I found a set of explicit solutions to $r_3^3$ and $r_3 r_5 r_6$ in $\mathbb{Q}(\sqrt{-7})$. We originally anticipated that, due to computing constraints, that we were going to have to find a finite field solution and lift it back to the number field using some Hensel's lifting tricks. However, I decided to just ask $\texttt{Magma}$ to find the irreducible components of the projective variety corresponding to the relations given by $r_3^3$ and $r_3 r_5 r_6$. Magma actually terminated in quite a reasonable time! So we saved all the effort with the Hensel lifting tricks.

With the equations of $r_3^3$ and $r_3 r_5 r_6$ and Yanxin's code producing the linear cuts for $3H + D$, I successfully produced an embedding of the fake projective plane $X = (a = 7, p = 2, \emptyset, D_3 2_7)$ into $\mathbb{C}P^5$.

I also calculated the explicit quadratics vanishing on $3H + D$, $3H + D_1$, and $3H + D + D_1$. This is a more refined set of equations defining each divisor.

During this week, I also presented my small project "Computing Flip Graphs of Highly Non-Convex Polygons" at DIMACS's Workshop on Modern Techniques in Graph Algorithms.

Week 4: 6/19-6/25

At the beginning of the week (on Sunday actually), Professor Borisov listed a series of goals he envisioned for this project:

  1. Compute equations of the unique sections of 3H + Torsion for all non-trivial torsion.
  2. Compute equations of 4H + Torsion for all non-trivial torsions and check if they have common zeros.
  3. Compute the equations of the embeddings of $X$ given by $5H$ and $5H + D$.
  4. In the family of fake projective planes $X = (a = 7, p = 2, \emptyset, D_3 2_7)$ belongs in, there's a closely related fake projective plane $Y = (a = 7, p = 2, \emptyset, D_3 X_7)$ whose algebraic equations haven't been explicitly computed before. We want to try to compute the equations for it. This is a mostly disjoint and quite ambitious topic from the others.
We have pretty much done the first goal and half of the third goal already. We decided to focus on the embedding in $5H$ is first. The plan is as follows, let $D_1, D_2, ..., D_7$ be the $C7$ orbit of $D_1$, then we can write \[30H = (5H) + (4H) + \sum_{i = 1}^7 (3H + D_i) \] Hence, we can find degree $5$ polynomials vanishing on $4H$ and $3H + D_i$ for all $i$ to find equations on $5H$, this will give us a basis of $6$ equations exactly, and this will be our map into $\mathbb{C}P^5$.

We went ahead and successfully found the map into $\mathbb{C}P^5$ using $5H$. Surprisingly, the surface we ended up with is actually singular! We double checked by finding quadratics vanishing on the $6$ sections of $5H$ that they don't have any common zeroes as well, so this is indeed a valid map. Furthermore, we suspect that the only singular points on the surface are given by $[1:0:0:0:0:0], [0:1:0:0:0:0],$ and $[0:0:0:1:0:0]$.

After finding the quadratics vanishing on $5H$. We were able to compute the quadratics vanishing on $4H + D$, $4H + D_1$, and $4H + D + D_1$ as follows - for each torsion $T$, we can write \[12H = (4H + T) + (3H + T) + 5H \] Hence we can find the $3$ sections on $4H + T$ as quadratic equations. Then we find points on $4H + T$ by finding zeroes of the quadratic equations that aren't on $3H + T$ or $5H$. With this, we calculated the quadratics vanishing on $4H + T$.

Week 5: 6/26-7/2

This week we started working towards goal $4$ of our project. The idea is as follows - the fake projective planes $X$ and $Y$ are both covered by a covering space $\mathbb{B}^2/G_4$. If we can lift the equations of $X$ up to $\mathbb{B}^2/G_4$, we can then find equations on $Y$. The covering over $X$ is a $14$-fold cover, broken into steps $\mathbb{B}^2/G_4 \to \mathbb{B}^2/G_5 \to X$.

The first covering is degree $7$ and the second is degree $2$. We first computed the equations of the bicanonical embedding of $\mathbb{B}^2/G_5$. This part is relatively straight forward. We first find $10$ sections for $6H + D$, given as cubic equations, then we use these $10$ sections to add $10$ more variables $U_{10}, ...., U_{19}$ on top of the original $10$ coordinates $U_0, ..., U_9$. Then we were able to produce $\mathbb{B}^2/G_5$ in coordinates $P_1, ..., P_20$ where \[P_1 = U_1, ..., P_9 = U_9, P_{10} = \frac{U_{10}}{\sqrt{U_{10}}}, ..., P_{20} = \frac{U_{20}}{\sqrt{U_{10}}}\] by satisfying certain even and odd relations.

The step of going from $\mathbb{B}^2/G_4$ to $\mathbb{B}^2/G_5$ is harder. The plan is to construct $13$ sections $z_0, ..., z_{12}$ to give an embedding of $\mathbb{B}^2/G_4$. $z_0, z_1, z_4, z_7, z_{10}$ can actually be calculated from $\mathbb{B}^2/G_5$, then we can compute the other sections using a $C3$ action (for more technical details, please see our upcoming paper about this). We started working ahead on computing these sections.

Week 6: 7/3-7/9

We want to the compute "a good multiplication table" describing how the products $z_i z_j$ are expressed in terms of the coordinates $P_i$. There should be enough free variables left in the end such that we can make the multiplication associative and scale for nice coefficients. By the end of the week, we finished computing a nice multiplication table.

With this multiplication table, the idea is to solve for coefficients $c_{i, j, k}$ on $\sum c_{i, j, k}\ z_i z_j z_k$ such that they are zero on $\mathbb{B}^2/G_5$. We ended up finding $66$ equations from this giving a map of $\mathbb{B}^2/G_5$ to $\mathbb{C}P^{12}$.

Unfortunately, this turns out to not be an embedding as there are singular points. We also checked that the sections $z_0, ..., z_{12}$ don't have any common zeroes, so this is a valid map.

Week 7: 7/10-7/16

Fortunately, there is a work around to this. On Tuesday morning, using a lot of non-singular points on this surface and building $10$ specific cubic sections in $z_i$, we are able to construct an embedding of $Y = (a = 7, p = 2, \emptyset, D_3 X_7)$ into $\mathbb{C}P^5$.

Some more simplications can be done, but for now the bulk of this project is done. The next steps are to write up the two projects we did into two papers.

I spent the rest of the week cleaning up all the code I wrote for the first project into a well-organized folder with detailed comments. I plan on cleaning up the code for the second project the following week.

Week 8: 7/17-7/23

This week I focused on writing up the first project we did and organizing the code repository while we are writing the paper. We also figured out a proof that $h^0(X, 2H + T) = 0$ for any torsion class $T \in Pic(X)$. I also gave a presentation on Friday about the outcomes of my work.

Week 9: 7/24-7/28

This week marks the end of my REU program. I finished my writeup of the first project and submitted it to the REU program. I also finished cleaning up the code for the second project finding equations for $Y = (a = 7, p = 2, \emptyset, D_3 X_7)$ and commented throughout the code. We plan on finishing up the writing of the second project in post REU.

References in this project

The following list is by no means exhaustive but serves as a guideline for references in this project.
  1. L. Borisov and J. Keum, Explicit equations of a fake projective plane, arXiv:1802.06333.
  2. L. Borisov and Z. Lihn, Realizing a Fake Projective Plane as a Degree 25 Surface in $\mathbb{P}^5$, arXiv:2301.09155.
  3. L. Borisov and E. Fatighenti, New explicit constructions of surfaces of general type, arXiv:2004.02637.
  4. D. Cartwright, T. Steger, Enumeration of the $50$ fake projective planes, C. R. Acad. Sci. Paris, Ser. I 348 (2010) 11-13.
  5. G. Prasad, S.-K. Yeung, Fake projective planes, Invent. Math. 168 (2007) 321-370; Addendum, 182 (2010) 213-227.
  6. D. Mumford. An algebraic surface with $K$ ample, $(K^2) = 9, p_g = q = 0$, American Journal of Mathematics, 101(1):233–244, 1979.
  7. Shing-Tung Yau. Calabi’s conjecture and some new results in algebraic geometry, Proceedings of the National Academy of Sciences of the United States of America, 74 5:1798–9, 1977.
  8. R. Hartshorne, Algebraic Geometry, Springer, New York, 1977.
  9. O. Forster, Lectures on Riemann Surfaces, Springer, New York, 1981.