Dear sage-devel members,
As you may have heard of, Robbie Hanson and myself have been working on a Sagemath package, CompGIT <https://github.com/Robbie-H/CompGIT/>, to carry out computations in geometric invariant theory (GIT) with Sage. We are writing to share the package with you and request any comments. Following the Sage Documentation <https://doc.sagemath.org/html/en/developer/github.html#planning-an-enhancement>, we would also like to ask your opinion on whether CompGIT would be of enough interest to open an issue/PR in the Sage repo for consideration to become a Sage enhancement. My initial motivation to write this in Sagemath was that I think it would be helpful to those people working with moduli spaces. Invariant theory, which GIT is a version of, also has applications to nearby fields, as detailed in Derksen-Kemper’s book <https://link.springer.com/book/10.1007/978-3-662-04958-7> and we are hoping the same will apply to CompGIT. Our package is based on algorithms and code prototypes <https://arxiv.org/abs/2308.08049> that I developed together with Patricio Gallardo, Han-Bom Moon and David Swinarski. Since we had never created a package for public use in the past, we would not have been able to complete this without the help and guidance of Frederic Chapoton, in particular with regards to conventions and doctesting. We have written a short manuscript <https://github.com/Robbie-H/CompGIT/blob/main/Documentation/CompGIT_documentation.pdf> that acts as a manual/list of potential applications. Nonetheless, we will quickly summarise the problem here, in perhaps a less technical language. We are happy to provide more detail as needed. In a nutshell, our library provides enough information to describe the points represented in the GIT quotient P(V)^{ss}//G where G is a simple group, V is a G-representation (a vector space with a G-action), P(V) is its projectivisation and P(V)^{ss}=P(V)\P(V)^{us} is the semistable locus, which excludes unstable points P(V)^{us} so that the quotient is an algebraic variety. What CompGIT does is telling you which G-orbits need to be excluded and which ones will represent orbits in the quotient (these are known as polystable orbits, which split into stable and strictly polystable). These mathematical notions go back to Hilbert but were formalised by Mumford in the 60s. They are a very powerful tool to give explicit descriptions of projective moduli of objects (such as algebraic curves). One of the applications of our work is to describe compactifications of the space of hypersurfaces in projective space. Due to Mumford’s general theory, in principle our library is enough to describe GIT quotients X//G where X<P(V) is a projective variety. In practice, this may be too difficult depending on the description one has of X inside P(V). Nonetheless, many examples of moduli are covered, including hypersurfaces, certain complete intersections or étale covers of either of those. We look forward to hearing from you. Best wishes, Robbie and Jesús *Jesús Martínez García* Senior Lecturer in Pure Mathematics School of Mathematics, Statistics and Actuarial Science University of Essex Office STEM 5.9 <https://findyourway.essex.ac.uk/bcdc98e0-e3c3-11eb-b52e-05a67b7792fc/search/projects/23/60ef1a882031e800c23040c4>, Colchester campus T +44 (0) 1206 873620 E jesus.martinez-gar...@essex.ac.uk (preferred), je...@jesusmartinezgarcia.net (alternative) ► www.jesusmartinezgarcia.net -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To view this discussion visit https://groups.google.com/d/msgid/sage-devel/802f74da-2051-42b4-9050-12a8ef950bd9n%40googlegroups.com.
