#11930: disallow non-smooth hyperelliptic curves, and let hyperelliptic curves
know
they are not singular
-------------------------------------------------------------+--------------
Reporter: dkrenn | Owner:
cremona
Type: defect | Status:
needs_review
Priority: major | Milestone:
sage-5.1
Component: elliptic curves | Resolution:
Keywords: hyperelliptic curve singular smooth sd35 | Work issues:
Report Upstream: N/A | Reviewers:
Marco Streng, Damiano Testa, David Eklund
Authors: Daniel Krenn, Marco Streng, Damiano Testa | Merged in:
Dependencies: | Stopgaps:
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Comment (by mstreng):
Replying to [comment:35 davideklund]:
> I only have two small comments on the patches:
I don't want the documentation to be confusing, so I'll address your
comments and post a new patch.
>
> About the example beginning with "A hyperelliptic curve should not be
given by polynomials of degree greater than `2g+2`, where `g` is the
genus." I'm not sure I understand what this means. The genus of what? It
is slightly unclear since I gather that in this example there is no
hyperelliptic curve whose genus we could be referring to. Is it the genus
of the desingularization of the corresponding plane curve?
Yes, I will explain that better.
> The way I view this, the equation {{{y^2+hy=f}}}, where {{{h=x^100}}}
and {{{f=x^6+1-h^2/4}}} (appropriately homogenized with {{{z}}}) defines a
curve in weighted projected space {{{P(1,100,1)}}} but that curve is
singular at the point {{{(x,y,z)=(1,-1/2,0)}}}.
Yes, and we should not allow that.
> About the example beginning "Input with integer coefficients creates
objects with the integers as base ring, but only checks smoothness over
`QQ`, not `ZZ`". Is this to be interpreted as saying that the example
provided is not smooth as a scheme over {{{Spec(ZZ)}}}?
Yes. But I see that Spec(ZZ) may confuse or scare people, so I will
clarify this.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11930#comment:36>
Sage <http://www.sagemath.org>
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