We provide computer-based solutions for the study of complex $K3$ surfaces.
Let $X$ be a complex algebraic $K3$ surface with Néron-Severi group $S=\text{NS}(X)$.
Study of the automorphism group of $X$,
Orbits of smooth rational curves
- Setting up the environment to execute Borcherds’ method.
- Can Borcherds’ method be applied to your K3 surface to obtain the automorphism group?
- Embedding update procedure.
- Running the Poolized Borcherds’ method to compute $\text{Aut}(X)$.
Denote by $\mathcal{P}_S$ the positive cone of $X$. Coming soon :
- Fundamental domain of the action of $\aut(X)$ onto $\text{Nef}(X) \cap \mathcal{P}_S$ and orbits of $(-2)$-curves.
- Toward a parallelized Borcherds’ method: Poolized Borcherds’ worker APFB.
NB : The Poolized Borcherds’ method enforces parallelism at the level of the internal procedures of the Borcherds’ method. When we write “towards a parallelized Borcherds’ method”, we aim for also enforcing parallelism at the level of the Borcherds’ method itself: That is, parallelizing the exploration & processing of the chamber structure, while remaining able to make use of parallelism within the method, i.e., at the level of its core components, like in our Poolized implementation of the Borcherds’ method which takes advantage of process-based parallelism.
Study of projective models of $X$
- Testing whether a class in the Néron-Severi group is ample : AmpTester
- Study of projective models with the SDM theorem : PModChecker, SysFinder & SysDisplay
MISC
- Sage port with Sage / Magma interface of Xavier Roulleau’s Magma implementation of an algorithm due to Vindberg.
- Three useful programs based on algorithms due to Shimada.
NB : Many other things (which are neither mentioned on this site nor in the manuscript) were done during the thesis, such as : the determination of explicit equations of $K3$ surfaces using a computer-based approach, asymptotic study of the distribution of classes of smooth rational curves (power model, log model, exp. model…), application of Merten’s algorithm to generate automorphisms of $K3$ surfaces.