Map to the Weierstrass form of a toric elliptic curve

There are 16 reflexive polygons in 2-d. Each defines a toric Fano variety, which (since it is 2-d) has a unique crepant resolution to a smooth toric surface. An anticanonical hypersurface defines a genus one curve \(C\) in this ambient space, with Jacobian elliptic curve \(J(C)\) which can be defined by the Weierstrass model \(y^2 = x^3 + f x + g\). The coefficients \(f\) and \(g\) can be computed with the weierstrass module. The purpose of this model is to give an explicit rational map \(C \to J(C)\). This is an \(n^2\)-cover, where \(n\) is the minimal multi-section of \(C\).

Since it is technically often easier to deal with polynomials than with fractions, we return the rational map in terms of homogeneous coordinates. That is, the ambient space for the Weierstrass model is the weighted projective space \(\mathbb{P}^2[2,3,1]\) with homogeneous coordinates \([X:Y:Z] = [\lambda^2 X, \lambda^3 Y, \lambda Z]\). The homogenized Weierstrass equation is

\[Y^2 = X^3 + f X Z^4 + g Z^6\]

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: cubic = x^3 + y^3 + 1
sage: f, g = WeierstrassForm(cubic);  (f,g)
(0, -27/4)

That is, this hypersurface \(C \in \mathbb{P}^2\) has a Weierstrass equation \(Y^2 = X^3 + 0 \cdot X Z^4 - \frac{27}{4} Z^6\) where \([X:Y:Z]\) are projective coordinates on \(\mathbb{P}^2[2,3,1]\). The form of the map \(C\to J(C)\) is:

sage: X,Y,Z = WeierstrassForm(cubic, transformation=True);  (X,Y,Z)
(-x^3*y^3 - x^3 - y^3,
 1/2*x^6*y^3 - 1/2*x^3*y^6 - 1/2*x^6 + 1/2*y^6 + 1/2*x^3 - 1/2*y^3,
 x*y)

Note that plugging in \([X:Y:Z]\) to the Weierstrass equation is a complicated polynomial, but contains the hypersurface equation as a factor:

sage: -Y^2 + X^3 + f*X*Z^4 + g*Z^6
-1/4*x^12*y^6 - 1/2*x^9*y^9 - 1/4*x^6*y^12 + 1/2*x^12*y^3
- 7/2*x^9*y^6 - 7/2*x^6*y^9 + 1/2*x^3*y^12 - 1/4*x^12 - 7/2*x^9*y^3
- 45/4*x^6*y^6 - 7/2*x^3*y^9 - 1/4*y^12 - 1/2*x^9 - 7/2*x^6*y^3
- 7/2*x^3*y^6 - 1/2*y^9 - 1/4*x^6 + 1/2*x^3*y^3 - 1/4*y^6
sage: cubic.divides(-Y^2 + X^3 + f*X*Z^4 + g*Z^6)
True

If you prefer you can also use homogeneous coordinates for \(C \in \mathbb{P}^2\)

sage: R.<x,y,z> = QQ[]
sage: cubic = x^3 + y^3 + z^3
sage: f, g = WeierstrassForm(cubic);  (f,g)
(0, -27/4)
sage: X,Y,Z = WeierstrassForm(cubic, transformation=True)
sage: cubic.divides(-Y^2 + X^3 + f*X*Z^4 + g*Z^6)
True

The 16 toric surfaces corresponding to the 16 reflexive polygons can all be blown down to \(\mathbb{P}^2\), \(\mathbb{P}^1\times\mathbb{P}^1\), or \(\mathbb{P}^{2}[1,1,2]\). Their (and hence in all 16 cases) anticanonical hypersurface can equally be brought into Weierstrass form. For example, here is an anticanonical hypersurface in \(\mathbb{P}^{2}[1,1,2]\)

sage: P2_112 = toric_varieties.P2_112()
sage: C = P2_112.anticanonical_hypersurface(coefficients=[1]*4);  C
Closed subscheme of 2-d CPR-Fano toric variety
covered by 3 affine patches defined by:
  z0^4 + z2^4 + z0*z1*z2 + z1^2
sage: eq = C.defining_polynomials()[0]
sage: f, g = WeierstrassForm(eq)
sage: X,Y,Z = WeierstrassForm(eq, transformation=True)
sage: (-Y^2 + X^3 + f*X*Z^4 + g*Z^6).reduce(C.defining_ideal())
0

Finally, you sometimes have to manually specify the variables to use. This is either because the equation is degenerate or because it contains additional variables that you want to treat as coefficients:

sage: R.<a, x,y,z> = QQ[]
sage: cubic = x^3 + y^3 + z^3 + a*x*y*z
sage: f, g = WeierstrassForm(cubic, variables=[x,y,z])
sage: X,Y,Z = WeierstrassForm(cubic, variables=[x,y,z], transformation=True)
sage: cubic.divides(-Y^2 + X^3 + f*X*Z^4 + g*Z^6)
True

REFERENCES:

• [AKMMMP2002]

sage.schemes.toric.weierstrass_covering.WeierstrassMap(polynomial, variables=None)

Return the Weierstrass form of an anticanonical hypersurface.

You should use sage.schemes.toric.weierstrass.WeierstrassForm() with transformation=True to get the transformation. This function is only for internal use.

INPUT:

• polynomial – a polynomial. The toric hypersurface equation. Can be either a cubic, a biquadric, or the hypersurface in \(\mathbb{P}^2[1,1,2]\). The equation need not be in any standard form, only its Newton polyhedron is used.

• variables – a list of variables of the parent polynomial ring or None (default). In the latter case, all variables are taken to be polynomial ring variables. If a subset of polynomial ring variables are given, the Weierstrass form is determined over the function field generated by the remaining variables.

OUTPUT:

A triple \((X,Y,Z)\) of polynomials defining a rational map of the toric hypersurface to its Weierstrass form in \(\mathbb{P}^2[2,3,1]\). That is, the triple satisfies

\[Y^2 = X^3 + f X Z^4 + g Z^6\]

when restricted to the toric hypersurface.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: cubic = x^3 + y^3 + z^3
sage: X,Y,Z = WeierstrassForm(cubic, transformation=True);  (X,Y,Z)
(-x^3*y^3 - x^3*z^3 - y^3*z^3,
 1/2*x^6*y^3 - 1/2*x^3*y^6 - 1/2*x^6*z^3 + 1/2*y^6*z^3
     + 1/2*x^3*z^6 - 1/2*y^3*z^6,
 x*y*z)
sage: f, g = WeierstrassForm(cubic);  (f,g)
(0, -27/4)
sage: cubic.divides(-Y^2 + X^3 + f*X*Z^4 + g*Z^6)
True

Only the affine span of the Newton polytope of the polynomial matters. For example:

sage: WeierstrassForm(cubic.subs(z=1), transformation=True)
(-x^3*y^3 - x^3 - y^3,
 1/2*x^6*y^3 - 1/2*x^3*y^6 - 1/2*x^6
     + 1/2*y^6 + 1/2*x^3 - 1/2*y^3,
 x*y)
sage: WeierstrassForm(x * cubic, transformation=True)
(-x^3*y^3 - x^3*z^3 - y^3*z^3,
 1/2*x^6*y^3 - 1/2*x^3*y^6 - 1/2*x^6*z^3 + 1/2*y^6*z^3
     + 1/2*x^3*z^6 - 1/2*y^3*z^6,
 x*y*z)

This allows you to work with either homogeneous or inhomogeneous variables. For example, here is the del Pezzo surface of degree 8:

sage: dP8 = toric_varieties.dP8()
sage: dP8.inject_variables()
Defining t, x, y, z
sage: WeierstrassForm(x*y^2 + y^2*z + t^2*x^3 + t^2*z^3, transformation=True)
(-1/27*t^4*x^6 - 2/27*t^4*x^5*z - 5/27*t^4*x^4*z^2
     - 8/27*t^4*x^3*z^3 - 5/27*t^4*x^2*z^4 - 2/27*t^4*x*z^5
     - 1/27*t^4*z^6 - 4/81*t^2*x^4*y^2 - 4/81*t^2*x^3*y^2*z
     - 4/81*t^2*x*y^2*z^3 - 4/81*t^2*y^2*z^4 - 2/81*x^2*y^4
     - 4/81*x*y^4*z - 2/81*y^4*z^2,
 0,
 1/3*t^2*x^2*z + 1/3*t^2*x*z^2 - 1/9*x*y^2 - 1/9*y^2*z)
sage: WeierstrassForm(x*y^2 + y^2 + x^3 + 1, transformation=True)
(-1/27*x^6 - 4/81*x^4*y^2 - 2/81*x^2*y^4 - 2/27*x^5
     - 4/81*x^3*y^2 - 4/81*x*y^4 - 5/27*x^4 - 2/81*y^4 - 8/27*x^3
     - 4/81*x*y^2 - 5/27*x^2 - 4/81*y^2 - 2/27*x - 1/27,
 0,
 -1/9*x*y^2 + 1/3*x^2 - 1/9*y^2 + 1/3*x)

By specifying only certain variables we can compute the Weierstrass form over the function field generated by the remaining variables. For example, here is a cubic over \(\QQ[a]\)

sage: R.<a, x,y,z> = QQ[]
sage: cubic = x^3 + a*y^3 + a^2*z^3
sage: WeierstrassForm(cubic, variables=[x,y,z], transformation=True)
(-a^9*y^3*z^3 - a^8*x^3*z^3 - a^7*x^3*y^3,
 -1/2*a^14*y^3*z^6 + 1/2*a^13*y^6*z^3 + 1/2*a^13*x^3*z^6
     - 1/2*a^11*x^3*y^6 - 1/2*a^11*x^6*z^3 + 1/2*a^10*x^6*y^3,
 a^3*x*y*z)

sage.schemes.toric.weierstrass_covering.WeierstrassMap_P1xP1(polynomial, variables=None)

Map an anticanonical hypersurface in \(\mathbb{P}^1 \times \mathbb{P}^1\) into Weierstrass form.

Input/output is the same as WeierstrassMap(), except that the input polynomial must be a standard anticanonical hypersurface in the toric surface \(\mathbb{P}^1 \times \mathbb{P}^1\):

EXAMPLES:

sage: from sage.schemes.toric.weierstrass_covering import WeierstrassMap_P1xP1
sage: from sage.schemes.toric.weierstrass import WeierstrassForm_P1xP1
sage: R.<x0,x1,y0,y1,a> = QQ[]
sage: biquadric = (x0^2*y0^2 + x1^2*y0^2 + x0^2*y1^2 + x1^2*y1^2 +
....:     a * x0*x1*y0*y1*5)
sage: f, g = WeierstrassForm_P1xP1(biquadric, [x0, x1, y0, y1]);  (f,g)
(-625/48*a^4 + 25/3*a^2 - 16/3, 15625/864*a^6 - 625/36*a^4 - 100/9*a^2 + 128/27)
sage: X, Y, Z = WeierstrassMap_P1xP1(biquadric, [x0, x1, y0, y1])
sage: (-Y^2 + X^3 + f*X*Z^4 + g*Z^6).reduce(R.ideal(biquadric))                 # needs sage.libs.singular
0

sage: R = PolynomialRing(QQ, 'x,y,s,t', order='lex')
sage: R.inject_variables()
Defining x, y, s, t
sage: equation = (s^2*(x^2+2*x*y+3*y^2) + s*t*(4*x^2+5*x*y+6*y^2)
....:             + t^2*(7*x^2+8*x*y+9*y^2))
sage: X, Y, Z = WeierstrassMap_P1xP1(equation, [x,y,s,t])
sage: f, g = WeierstrassForm_P1xP1(equation, variables=[x,y,s,t])
sage: (-Y^2 + X^3 + f*X*Z^4 + g*Z^6).reduce(R.ideal(equation))                  # needs sage.libs.singular
0

sage: R = PolynomialRing(QQ, 'x,s', order='lex')
sage: R.inject_variables()
Defining x, s
sage: equation = s^2*(x^2+2*x+3) + s*(4*x^2+5*x+6) + (7*x^2+8*x+9)
sage: X, Y, Z = WeierstrassMap_P1xP1(equation)
sage: f, g = WeierstrassForm_P1xP1(equation)
sage: (-Y^2 + X^3 + f*X*Z^4 + g*Z^6).reduce(R.ideal(equation))                  # needs sage.libs.singular
0

sage.schemes.toric.weierstrass_covering.WeierstrassMap_P2(polynomial, variables=None)

Map a cubic to its Weierstrass form.

Input/output is the same as WeierstrassMap(), except that the input polynomial must be a cubic in \(\mathbb{P}^2\),

\[\begin{split}\begin{split} p(x,y) =&\; a_{30} x^{3} + a_{21} x^{2} y + a_{12} x y^{2} + a_{03} y^{3} + a_{20} x^{2} + \\ &\; a_{11} x y + a_{02} y^{2} + a_{10} x + a_{01} y + a_{00} \end{split}\end{split}\]

EXAMPLES:

sage: from sage.schemes.toric.weierstrass import WeierstrassForm_P2
sage: from sage.schemes.toric.weierstrass_covering import WeierstrassMap_P2
sage: R.<x,y,z> = QQ[]
sage: equation =  x^3 + y^3 + z^3 + x*y*z
sage: f, g = WeierstrassForm_P2(equation)
sage: X,Y,Z = WeierstrassMap_P2(equation)
sage: equation.divides(-Y^2 + X^3 + f*X*Z^4 + g*Z^6)
True

sage: from sage.schemes.toric.weierstrass import WeierstrassForm_P2
sage: from sage.schemes.toric.weierstrass_covering import WeierstrassMap_P2
sage: R.<x,y> = QQ[]
sage: equation =  x^3 + y^3 + 1
sage: f, g = WeierstrassForm_P2(equation)
sage: X,Y,Z = WeierstrassMap_P2(equation)
sage: equation.divides(-Y^2 + X^3 + f*X*Z^4 + g*Z^6)
True

sage.schemes.toric.weierstrass_covering.WeierstrassMap_P2_112(polynomial, variables=None)

Map an anticanonical hypersurface in \(\mathbb{P}^2[1,1,2]\) into Weierstrass form.

Input/output is the same as WeierstrassMap(), except that the input polynomial must be a standard anticanonical hypersurface in weighted projective space \(\mathbb{P}^2[1,1,2]\):

\[\begin{split}\begin{split} p(x,y) =&\; a_{40} x^4 + a_{30} x^3 + a_{21} x^2 y + a_{20} x^2 + \\ &\; a_{11} x y + a_{02} y^2 + a_{10} x + a_{01} y + a_{00} \end{split}\end{split}\]

EXAMPLES:

sage: from sage.schemes.toric.weierstrass_covering import WeierstrassMap_P2_112
sage: from sage.schemes.toric.weierstrass import WeierstrassForm_P2_112
sage: R = PolynomialRing(QQ, 'x,y,a0,a1,a2,a3,a4', order='lex')
sage: R.inject_variables()
Defining x, y, a0, a1, a2, a3, a4
sage: equation = y^2 + a0*x^4 + 4*a1*x^3 + 6*a2*x^2 + 4*a3*x + a4
sage: X, Y, Z = WeierstrassMap_P2_112(equation, [x,y])
sage: f, g = WeierstrassForm_P2_112(equation, variables=[x,y])
sage: (-Y^2 + X^3 + f*X*Z^4 + g*Z^6).reduce(R.ideal(equation))                  # needs sage.libs.singular
0

Another example, this time in homogeneous coordinates:

sage: fan = Fan(rays=[(1,0),(0,1),(-1,-2),(0,-1)],
....:           cones=[[0,1],[1,2],[2,3],[3,0]])
sage: P112.<x,y,z,t> = ToricVariety(fan)
sage: (-P112.K()).sections_monomials()
(z^4*t^2, x*z^3*t^2, x^2*z^2*t^2, x^3*z*t^2,
 x^4*t^2, y*z^2*t, x*y*z*t, x^2*y*t, y^2)
sage: C_eqn = sum(_)
sage: C = P112.subscheme(C_eqn)
sage: WeierstrassForm_P2_112(C_eqn, [x,y,z,t])
(-97/48, 17/864)
sage: X, Y, Z = WeierstrassMap_P2_112(C_eqn, [x,y,z,t])
sage: (-Y^2 + X^3 - 97/48*X*Z^4 + 17/864*Z^6).reduce(C.defining_ideal())        # needs sage.libs.singular
0