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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