Script final

I want to thank everyone in the audience, whether in this room or online, for attending my PhD thesis defense and thank to the jury, Madame Sarti, Mister Mangolte, Mister Lairez, and huge thanks to Alice Garbagnati and Davide Cesare Veniani, reviewers of this thesis. 

I also want to thank my advisor Xavier Roulleau without whom none of this would have been possible.

My thesis is entitled a computer-based algorithmic approach to the study of $K3$ surfaces. Besides a brief introduction, during which I will present, the main object of study (i.e., K3 surfaces), the goals of this thesis, the means implemented to reach these goals, and the framework within which this thesis fits as a development,

this presentation has been structured around two axes.

  • One focuses on the automorphism aspect of this thesis,
  • the other part focuses on basics of the parallel computing aspect

So, let’s set things up let’s set things up

Compact complex surfaces with trivial canonical divisor and irregularity equal to one are called K3 surfaces.

K3 surfaces constitute a central object of study in pure mathematics and form a fertile ground for the application of many theoretical tools. As indicated by Simion Filip, K3 surfaces provide a meeting ground for algebraic geometry, differential geometry, arithmetic, and dynamics.

One point that is of particular interest for us and that it not mentioned in this citation is that K3 surfaces can be defined in the framework of lattice theory. It is indeed well known that the second cohomology group with integers coefficients of a $K3$ surface is isomorphic to the direct sum of three copies of $U$ and two copies of $E_{8}(-1)$, thus endowing this cohomology group with a lattice structure, called the $K3$ lattice.

A fundamental theorem in the theory of $K3$ surfaces states that for any primitive hyperbolic sublattice S of the $K3$ lattice, there exists a $K3$ surface $X$ such that $S$ is isomorphic to the Néron-Severi group of X.

We recall that the Néron-Severi group of a variety is the group of divisors modulo algebraic equivalence and that this group is isomorphic to the Picard lattice in the case of K3 surfaces (exponential exact sequence).

With these considerations in mind, we can now introduce the goals defined by our advisor Xavier Roulleau that we had to complete to achieve this doctoral project.

We were tasked with the study of K3 surfaces, X_t, having Néron-Severi group isomorphic to the lattice with Gram matrix of size 3×3, diagonal coefficients 2t, -2, -2, non-diagonal coefficients zero, for an integer parameter t, say contained in the interval 1,2,3, 4, 5, …, forty-eight, forty-nine, fifty.

In particular, for each such value of t, we had to compute the Automorphism Groups of these surfaces, and find, an upper bound on the number of orbits of classes of smooth rational curves under the action of $\aut(X_t).

Standing in front of this audience, I am pleased to announce today that, following this thesis, all this can be done with full automation, without having to break a sweat.

To achieve these goals, we used a computer-based algorithmic approach and ended up with tools having a scope of application and purpose going way beyond what had been initially envisioned during the early days of this thesis.

With a little demonstration, let me show you that :

  1. All the objectives defined to achieve this thesis have been met.
  2. The scope of application of the tools used to do so is indeed very wide.

It won’t take more than a minute.

So, here are to $K3$ surfaces.

This strategy consists in implementing programs in such a way that all steps that would normally have to be taken to study K3 surfaces, for instance, computing automorphism groups, orbits of smooth rational curves, projective models, the fundamental domain of the action of the automorphism group onto the nef cone, unirationality, equations, can be fully automatized. 

Our first contact with this way of doing things dates back to two thousand nineteen. Xavier Roulleau produced a Magma program SmoothRationalCurves, released along his twenty nineteen article.


i have to mention that

Thus, we see that this thesis ended up at the interface

A decade ago, Ichiro Shimada, from Hiroshima University, tried to launch a new trend: The computational study of K3 surfaces, in the broader framework of the computational analysis of algebraic geometry.

In particular, Professor Shimada wrote two articles that significantly impacted this thesis.

 An algorithm to compute automorphism groups of $K3$ surfaces
and an application to singular $K3$ surfaces


Projective models of the supersingular $K3$ surface with Artin invariant 1 in characteristic $5$

Shimada’s 2013 article introduces an Algorithmic method, called the Borcherds’ method, to compute a generating set of the automorphism group of a complex $K3$ surface.

Shimada’s 2012 article provides guidelines regarding a lot of algorithmic material, in particular, a short lattice vectors enumerator, among other things, with HUGE, really massive UNTAPPED potential. AmpTester, PModChecker.

Since $2013$, Shimada’s material has neither been picked up by other researchers nor developed by Shimada himself.

That is, The trend initiated by Shimada, that is, the computer-based study of $K3$ surfaces, seems to have been at a standstill for many years, and often surrounded by some form of confusion.

Giacomo Mezzedimi, in his PhD thesis defended in late 2021, indicates, when referring to Shimada’s twenty-thirteen article on automorphisms, that

“Shimada presents an algorithm to compute the automorphism group of these K3 surfaces; however the full automorphism group can only be computed for a finite number of Picard lattices”

Giacomo Mezzedimi, PhD thesis

implying that the scope of application of Shimada’s material is limited to particular cases, and is clearly finite.

Shimada’s algorithm in fact opens the door to an infinite number of possibilities of study.

However, no one can blame Mezzedimi for his statement: Shimada’s article. as we indicate in the introduction to this thesis, Shimada’s article was not generalist and focused on examples without explicitly highlighting a general application framework for Borcherds’ method.

 In addition, many grey areas surrounded the steps that have to be taken to implement essential procedures described in Shimada’s article.

 Various fundamental aspects crucial to the generalization, optimization, and implementation of the processes were often ignored or treated in a minimalist way.

From an industrial, automated, and generalist point of view, Shimada’s groundbreaking article offered considerable possibilities for continuation: Shimada provides building blocks, guidelines, and hints regarding various procedures. His article is a bit like a puzzle. We modernized, enhanced, generalized and built all the missing pieces that we identified and even We opened a bridgehead towards future developments, by adding a modern and trendy flavor to Shimada’s material by making it ready for parallel/cloud computing on various levels.

We produced user-friendly solutions that can be used as a basis by other researchers.

Will release publicly all the programs produced during this thesis after the defense.

Before proceeding further, we introduce the main notations used from now until the end of this presentation.

We will now explain the mechanics behind Shimada’s vision regarding the Borcherds’ method. 

Whenever the complex $K3$ surface under study has a Picard number strictly less than $20$ and satisfies a condition that we call the Kernel condition, will return a generating set of $\aut(X)$.

The idea is that, when the surface satisfies certain conditions, then general theory ensures that all the information regarding the automorphism group of $X$ can be obtained from special transformations preserving a chamber structure on the intersection of the Nef cone of $X$ with its positive cone.

The purpose of the Borcherds’ method consists in retrieving this information, to finally return a generating set of $\aut(X)$. We are here dealing with a brute force procedure. Generators are obtained by BRUTE FORCE search, i.e., exhaustive search. More precisely, the method explores and processes the chamber structure over $\nps$ until a complete set of representatives

Using basic logic, we derived a generalized criterion from Shimada’s 2013 article to identify generators of $\aut(X)$.

More precisely, we provide a generalized criterion to identify transformation of a group $\hbf$ such that $$\aut_{\hbf}(\nps) \simeq \aut(X). $$

Using this criterion, BM the Borcherds’ method explores and processes the chamber structure to enumerate all the potential automorphisms.

The goal consists in obtaining a complete set of representatives of congruence classes of chambers contained in $\nps$.

As indicated by the name, the chamber structure over $\nps$ is the scene where the action takes place.

It is made of chambers. First, let’s explain how to set up a chamber structure on $\nps$.

The Néron-Severi group of X is assumed to be embedded primitively into an even hyperbolic lattice. The latter is chosen according to the Picard number of X, as specified in the lattice table A chamber structure on $\pl$ is obtained by taking the collection of closures of connected components of the intersection 

$$ \bigcap_{v\in \rl} \pl \; \backslash \; (v)^{\perp}.$$

How can we obtain a chamber structure on $\ps$?

We are now going to understand why we had to embed $S$ into $\LL$. The chamber structure on $\ps$ is obtained from the chamber structure on $\pl$.

The collection of intersections of $\pl$-chambers with $\ps$, that have the property of having a non-empty interior, is a chamber structure on $\ps$.

A $\pl$-chamber is said to be $\iota(S)$-nondegenerate whenever it induces a $\ps$-chamber contained in $\nps$.

Once this $\ps$-chamber structure is defined, we just have to introduce the notion of $(-2)$-walls to be able to define natural chamber structure over $\nps$.

As we indicated at the beginning of this section, the BM explores and processes the chamber structure over $\nps$ in order to fulfill its mission: computing a generating set of $\aut(X)$.

Thus, we have to make sure that the BM starts its exploration from within $\nps$ never leaves the chamber structure over $\nps$.

Providing the Method with the Weyl vector of an initial chamber contained in $\nps$ is ESSENTIAL.

Finding a chamber having such properties is, however, very hard.

We have 2 tools, which, when combined, enable us to deal with this issue :

Non-degeneracy test : DegenTest

The embedding update procedure.

The following little “animation” enables us to understand what happens when the BM is not provided with an initial chamber contained in $\nps$.