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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.