Using Professor Shimada’s work on automorphism groups of K3 surfaces as a starting point, we produced a generalized and fully automated implementation of the Borcherds’ method. This implementation uses process-based parallelism to compute a generating set of $\aut(X)$, a set containing representatives of all the orbits of smooth rational curves on $X$ and much more.
The only input data required consists of a Gram matrix of $S=\text{NS}(X)$, of an ample class, and vectors defining a primitive embedding of $S$ into a suitable ambient even hyperbolic lattice, as explained in the first point below.
The Borcherds’ method
- Setting up the environment to execute the Borcherds’ method.
- Can the Borcherds’ method be applied to your K3 surface to obtain the automorphism group?
- Embedding update procedure.
- Computing $\text{Aut}(X)$ : Running the Poolized Borcherds’ method.
Parallelism & The Borcherds’ method
- Process-based parallelism – Borcherds’ method & Pool
- Deploying the Poolized Borcherds’ method in parallel with PFB / APFB – Part 1
- Deploying the Poolized Borcherds’ method in parallel with PFB / APFB – Part 2
- Enforcing the PFB / APFB strategy for parallel deployment at the network level
Coming soon :
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. We fulfill this goal by enforcing our PFB / APFB strategy.