Nils, thanks to you, too, for your responses. On Saturday, October 28, 2023 at 11:16:39 AM UTC-7 Nils Bruin wrote:
> On Saturday, 28 October 2023 at 05:39:26 UTC-7 Kwankyu wrote: > > I looked the Magma code in ask.sagemath. There's no problem in computing a > canonical divisor for the curve (through the attached function field). > Computing a basis of the Riemann-Roch space is no problem as well. Actually > the hard part is to construct the morphism from C to P2 from the basis. > Magma does this seamlessly. But Sage lacks this functionality (perhaps > because I did not implement it). I think, the gist of the matter is to > convert an element of the function field to a rational function of the > coordinate ring of P2. > > > That's actually trivially simple: if [f1,f2,f3] is the basis of your > Riemann-Roch space, you just consider the map defined by [f1:f2:f3]. So you > lift f1,f2,f3 to rational functions on the affine space that contains your > curve: you just take the rational function representation and forget the > algebraic relations between the variables. You now have rational functions > in a rational function field, so you can clear denominators there. Now you > have a rational map (described by polynomials) A^2->P^r under which the > rational image of your curve C in A^2 is the corresponding projective > image. Computing that image is the usual groebner-basis operation for > finding images of rational maps, so that's potentially quite expensive. > > In practice, you know something about the denominators of the > representations of f1,f2,f3, so you can probably do a little better. > > At its core, that is what the magma code does too, although perhaps it has > some smart tricks here and there to try and keep degrees in check a bit. > -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-support/9449eaa0-6470-4cd2-91a5-6780f1faab86n%40googlegroups.com.
