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# Why am I getting a type error when I attempt to take the projective closure of this intersection?

I am attempting to take the projective closure of the intersection of the following affine polynomials (not the intersection of the closure!):

$$y^3-y-x^2= 0$$ $$w^3-w+y^7-y^5-x^4y^3+x^4y = 0.$$

This affine intersection is a curve of dimension one. Unfortunately, I haven't been able to enter this lovely affine curve defined by this intersection into sage. When I attempt as follows:

A.<x,y,w> = AffineSpace(QQ, 3)
P.<u,v,t,s>=ProjectiveSpace(QQ,3)
C = Curve([y^3-y-x^2, w^3-w+y^7-y^5-x^4*y^3+x^4*y], A)
D=C.projective_closure(1,P)


I get an error at the definition of C due to the second polynomial:

TypeError:  F (=[-x^4*y^3 + y^7 + x^4*y - y^5 + w^3 - w]) must be a list or tuple of polynomials of the coordinate ring of A (=Affine Space of dimension 3 over Finite Field of size 3)


I am so confused because this is absolutely in the coordinate ring of $A$. Why am I getting this type error? How can I enter this affine intersection into sage, so that I may take its closure?

As an aside, I can enter the intersection of the closure, which is not what I want, as follows:

x,y,z,w = GF(3)['x,y,z,w'].gens()
C = Curve([y^3-y*z^2-x^2*z, w^3*z^4-w*z^6+y^7-y^5*z^2-x^4*y^3+x^4*y*z^2]); C


The intersection of the closure has an extra irreducible component $[x: 0: w: 0]$,

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## 1 Answer

Sort by » oldest newest most voted The code seems to work as given over here. Have you tried it in the latest version of Sage?

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## Comments

I can confirm this works for me in Sage 9.2 and Sage 9.3.rc0:

sage: A.<x, y, w> = AffineSpace(QQ, 3)
sage: P.<u, v, t, s> = ProjectiveSpace(QQ, 3)
sage: C = Curve([y^3 - y - x^2, w^3 - w + y^7 - y^5 - x^4*y^3 + x^4*y], A)
sage: %time D = C.projective_closure(1, P)
CPU times: user 44.8 s, sys: 810 ms, total: 45.6 s
Wall time: 47.7 s


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Asked: 2021-04-01 05:08:59 +0200

Seen: 35 times

Last updated: Apr 01