Defining a general curve

asked 2013-03-22 12:48:07 +0100

Hey, I am relatively new to Sage, so this question might be very simple: I want to define a projective curve in $\mathbb{P}^2$ given by $x^3=y^3-Az^3$, where $A$ is an arbitrary non-zero element of an algebraically closed field, say C. I tried the following:

x,y,z=ProjectiveSpace(2,CC,'xyz').gens();

var('A',domain=CC);

C=Curve(x^3+y^3-A*z^3);

But I get TypeError: F (=-A*z^3 + x^3 + y^3) must be a multivariate polynomial.

Is it possible to define the curve I want in Sage? All examples of algebraic geometry in Sage I saw deal with explicitly defined curves.

answered 2013-03-26 16:29:14 +0100

lftabera
171 ●4 ●6

In this case A would be a variable. You might do the following

sage: K=QQ['A'].fraction_field()
sage: K.inject_variables()
Defining A
sage: R=K['x,y,z']
sage: R.inject_variables()
Defining x, y, z
sage: F=-A*z^3 + x^3 + y^3
sage: Curve(F)
Projective Curve over Fraction Field of Univariate Polynomial Ring in A over Rational Field defined by x^3 + y^3 + (-A)*z^3

But I am not sure if this is what you really want. I am afraid that curves defined over transcendental fields will have few working methods.

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Thank you! In the meanwhile I had figured that out. There is also the command K. = FunctionField(QQ). However you are right, the functionality is limited and it would be much nicer to work over QQ. I'm not sure if this is possible.

Michalis N ( 2013-03-27 11:51:23 +0100 )edit

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