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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 E8(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.

LAUNCH DEMO.

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

and 

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 Nef(X)PS 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 H such that AutH(Nef(X)PS)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 Nef(X)PS.

As indicated by the name, the chamber structure over Nef(X)PS 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 Nef(X)PS.

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 

vRLPL(v).

How can we obtain a chamber structure on PS?

We are now going to understand why we had to embed S into L. 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 ι(S)-nondegenerate whenever it induces a PS-chamber contained in Nef(X)PS.

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 Nef(X)PS.

As we indicated at the beginning of this section, the BM explores and processes the chamber structure over Nef(X)PS 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 Nef(X)PS never leaves the chamber structure over Nef(X)PS.

Providing the Method with the Weyl vector of an initial chamber contained in Nef(X)PS 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 Nef(X)PS.