Thanks for all of your posts, Kwankyu. Helpful and informative. John
On Saturday, October 28, 2023 at 6:19:48 AM UTC-7 Kwankyu wrote: > To answer John's question: > > sage: P2.<x,y,z> = ProjectiveSpace(QQ, 2) > sage: f = 2*x^5 - 4*x^3*y*z + x^2*y*z^2 + 2*x*y^3*z + 2*x*y^2*z^2+ y^5 > sage: C = P2.curve(f) > sage: F = C.function_field() > sage: z, = F.gens() > sage: K = z.differential().divisor() # canonical divisor > sage: (-K).dimension() > 3 > sage: f1, f2, f3 = (-K).basis_function_space() > sage: phi = C.hom(P2, [f1,f2,f3]). <--------------- does not work > sage: phi.image() # will work > > On Saturday, October 28, 2023 at 9:59:58 PM UTC+9 Kwankyu wrote: > >> Let me mention also the related PR >> >> https://github.com/sagemath/sage/pull/35467 >> >> which implements Jacobian groups of curves (again via function field), >> referencing Nils' old code. The PR is long sleeping in draft state. If >> anyone finds it useful, I may wake it up. >> >> On Saturday, October 28, 2023 at 9:39:26 PM UTC+9 Kwankyu wrote: >> >>> Hi, >>> >>> I replied to Dima's comment in >>> https://github.com/sagemath/sage/commit/977ace651af9b99689f7b6507f91f8b4e2588ae9#r131138149 >>> . >>> >>> Note that the "divisor" method of a curve had existed long before I >>> added function field machinery and attached function fields to curves. >>> Hence actually there are two systems of "divisors" of curves in Sage. >>> >>> The old system was implemented by William Stein, David Kohel, and Volker >>> Braun. In the old system, a divisor is a formal sum of rational points with >>> multiplicities. It is mainly implemented in >>> `src/sage/schemes/generic/divisor.py`. Overall it is very rudimentary. Dima >>> and John is attempting to use this system. >>> >>> The new system was implemented by me. Here a divisor is a formal sum of >>> places of a function field with multiplicities. This system is available >>> via the function field attached to a curve. This is much more powerful than >>> the old system. You can compute the Riemann-Roch space of a divisor. Nils >>> is using this system. >>> >>> I never attempted to combine the two systems, being afraid of breaking >>> the old system (or just being lazy :-) There are similarly two systems in >>> Magma too. But in Magma, the two systems are integrated tightly and >>> seamlessly. I did some integration in Sage too but far from complete >>> compared with Magma. >>> >>> 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. I have no idea how to do this now... Once you >>> construct the morphism, Sage can also compute the image of the morphism >>> (perhaps I implemented this). Hence unfortunately the Magma code cannot be >>> line by line converted to Sage code at present. >>> >>> On Saturday, October 28, 2023 at 8:27:07 AM UTC+9 Dima Pasechnik wrote: >>> >>>> On Sat, Oct 28, 2023 at 1:02 AM John H Palmieri <jhpalm...@gmail.com> >>>> wrote: >>>> >>>> > Yes, I noticed that, too. It also fails to provide any information >>>> about what ``v`` should be (beyond saying that it should be a "valid >>>> object"): there is no INPUT block. >>>> >>>> I've left a comment here: >>>> >>>> https://github.com/sagemath/sage/commit/977ace651af9b99689f7b6507f91f8b4e2588ae9#r131117132 >>>> >>>> >>>> fortunately, the author, @kwankyu is active >>>> >>>> I can't locate the ticket, but it was merged in 9.0.beta9 >>>> >>>> >>>> > >>>> > >>>> > On Friday, October 27, 2023 at 3:51:10 PM UTC-7 Dima Pasechnik wrote: >>>> >> >>>> >> By the way, the docstring of divisor() misses an example, it's >>>> >> >>>> >> def divisor(self, v, base_ring=None, check=True, reduce=True): >>>> >> r""" >>>> >> Return the divisor specified by ``v``. >>>> >> >>>> >> .. WARNING:: >>>> >> >>>> >> The coefficients of the divisor must be in the base ring >>>> >> and the terms must be reduced. If you set ``check=False`` >>>> >> and/or ``reduce=False`` it is your responsibility to pass >>>> >> a valid object ``v``. >>>> >> >>>> >> EXAMPLES:: >>>> >> >>>> >> sage: x,y,z = PolynomialRing(QQ, 3, names='x,y,z').gens() >>>> >> sage: C = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2) >>>> >> >>>> >> """ >>>> >> >>>> >> Is there an issue for this? >>>> >> >>>> >> On Sat, Oct 28, 2023 at 12:42 AM Nils Bruin <nbr...@sfu.ca> wrote: >>>> >> > >>>> >> > A canonical divisor is the divisor of any differential on C so the >>>> following does the trick: >>>> >> > >>>> >> > sage: kC=C.function_field() >>>> >> > sage: kC(kC.base_field().gen(0)).differential().divisor() >>>> >> > >>>> >> > It doesn't look like we quite have computation of Riemann-Roch >>>> spaces natively in sage yet, so finding effective representatives requires >>>> a little more work. In the RiemannSurface code this is done using >>>> singular's adjoint ideal code (or by Baker's theorem in cases where it >>>> applies). For this curve the canonical class is of degree -2, so there are >>>> no effective representatives in this case. >>>> >> > >>>> >> > On Friday, 27 October 2023 at 15:14:00 UTC-7 John H Palmieri >>>> wrote: >>>> >> >> >>>> >> >> If anyone here knows anything about canonical divisors and their >>>> implementation in Sage, please see >>>> https://ask.sagemath.org/question/74034/converting-algebraic-geometry-magmas-code-to-sage/. >>>> >>>> The setup: >>>> >> >> >>>> >> >> sage: P2.<x,y,z> = ProjectiveSpace(QQ, 2) >>>> >> >> sage: f = 2*x^5 - 4*x^3*y*z + x^2*y*z^2 + 2*x*y^3*z + >>>> 2*x*y^2*z^2+ y^5 >>>> >> >> sage: C = P2.curve(f) >>>> >> >> >>>> >> >> How do you get the canonical divisor for C? >>>> >> >> >>>> >> >> (I encourage you to post answers directly to ask.sagemath.org, >>>> if you're willing.) >>>> >> >> >>>> >> >> -- >>>> >> >> John >>>> >> >> >>>> >> > -- >>>> >> > 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...@googlegroups.com. >>>> >> > To view this discussion on the web visit >>>> https://groups.google.com/d/msgid/sage-support/91b14570-b83e-4dbf-8bca-0a2eff538a50n%40googlegroups.com. >>>> >>>> >>>> > >>>> > -- >>>> > 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...@googlegroups.com. >>>> > To view this discussion on the web visit >>>> https://groups.google.com/d/msgid/sage-support/391d8ee7-0329-4a15-bc88-4b84973389abn%40googlegroups.com. >>>> >>>> >>>> >>> -- You received this message because you are subscribed to the Google Groups "sage-support" group. 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