|
We have two strategies for computing the worst 1-PS for this curve:
Approach 1 We compute the Chow form \(R_C\) following Dalbec and Sturmfels, Section 1.2 and Gelfand, Kapranov, and Zelevinsky, Chapter 2 Section 5B and Example 4.8. (See the References section of our paper for the full citation details.) Then we compute the Chow polytope and the worst 1-PS. Let \(L\) be a line in \(\mathbb{P}^3\). We can write \(L\) as \[ \ker \left[ \begin{array}{cccc} u_0 & u_1 & u_2 & u_3 \\ v_0 & v_1 & v_2 & v_3 \end{array} \right] \] Then \(L\) meets \(C\) if and only if \begin{align*} u_0 s^4 + u_1 s^2 + u_2 s + u_3 &= 0 \\ v_0 s^4 + v_1 s^3 + v_2 s + v_3 &= 0 \end{align*} Recall: two polynomials \(f\) and \(g\) have a common zero if and only if their resultant vanishes. In this context, there are several resultants (the Sylvester resultant, the Bézout resultant, the Macaulay resultant,...). We use the Bézout resultant, as it leads most directly to the desired result. Write \begin{align*} f & = u_0 s^4 + u_1 s^2 + u_2 s + u_3 \\ g & = v_0 s^4 + v_1 s^3 + v_2 s + v_3 \end{align*} The Bézout resultant of \(f\) and \(g\) is the determinant of the matrix \(B\) defined by \[ \frac{f(x)g(y)-f(y)g(x)}{x-y} = \sum b_{i,j}x^i y^j \] We get \begin{align*} R_C & = \text{Béz}(f,g) \\ & = \det \left[ \begin{array}{cccc} p_{2,3} & p_{1,3} & 0 & p_{0,3} \\ p_{1,3} & p_{1,2} & p_{0,3} & p_{0,2} \\ 0 & p_{0,3} & p_{0,2} & p_{0,1} \\ p_{0,3} & p_{0,2} & p_{0,1} & 0 \end{array}\right]\\ &= p_{0,3}^4 -p_{0,2} p_{0,3}^2 p_{1,2}+2 p_{0,2}^2 p_{0,3} p_{1,3}-2 p_{0,1} p_{0,3}^2 p_{1,3}\\ & \quad \mbox{}+p_{0,1}^2 p_{1,3}^2 -p_{0,2}^3 p_{2,3} +2 p_{0,1} p_{0,2} p_{0,3} p_{2,3} -p_{0,1}^2 p_{1,2} p_{2,3}, \end{align*} where \(p_{ij} = u_i v_j - u_j v_i\) are the Plücker coordinates on \(\operatorname{Gr}(2,4)\).
Next, we compute the character of the maximal torus \(T\) acting on each monomial in \(R_C\). \[ \begin{array}{lll} \text{Monomial} & \quad & \text{Character} \\ p_{0,3}^4 & \qquad & (4,0,0,4) \\ p_{0,2} p_{0,3}^2 p_{1,2} & \qquad & (3,1,2,2)\\ p_{0,2}^2 p_{0,3} p_{1,3}& \qquad & (3,1,2,2) \\ p_{0,1} p_{0,3}^2 p_{1,3}& \qquad & (3,2,0,3)\\ p_{0,1}^2 p_{1,3}^2 & \qquad & (2,4,0,2) \\ p_{0,2}^3 p_{2,3}& \qquad & (3,0,4,1) \\ p_{0,1} p_{0,2} p_{0,3} p_{2,3}& \qquad & (3,1,2,2) \\ p_{0,1}^2 p_{1,2} p_{2,3}& \qquad & (2,3,2,1) \end{array} \] Thus, the Chow polytope of \( C\) is the convex hull of five points, \( (2,3,2,1), (3,0,4,1), (2,4,0,2), (4,0,0,4), (3,1,2,2)\). We find that the Chow polytope of \(C\) has four vertices; the point \( (3,1,2,2)\) is contained in the convex hull of the other four points.
The trivial character \( \chi_0\) is \( (2,2,2,2)\). We find that \( \chi_0 \not\in \operatorname{Chow}(C)\). The proximum \(p \in \operatorname{Chow}(C)\) to \( \chi_0\) is the point \( (\frac{88}{35},\frac{72}{35},\frac{64}{35},\frac{56}{35})\). We obtain \( (9,1,-3,-7) = \frac{35}{2}( (\frac{88}{35},\frac{72}{35},\frac{64}{35},\frac{56}{35}) - (2,2,2,2 ) \) after subtracting and scaling. Thus, \( \lambda(t) = (t^9,t,t^{-3},t^{-7})\) is a worst 1-PS for the Chow point of \(C\).
Approach 2 We compute the Chow polytope as the limit of the Hilbert state polytopes following Kapranov, Sturmfels, and Zelevinsky. Recall that \( C\) was defined by the parametrization \(t \mapsto (1,t^2,t^3,t^4)\). First, we eliminate the parameter to obtain the ideal of \(C\).
Macaulay2, version 1.24.05
with packages: ConwayPolynomials, Elimination, IntegralClosure,
InverseSystems, Isomorphism, LLLBases, MinimalPrimes,
OnlineLookup, PrimaryDecomposition, ReesAlgebra,
Saturation, TangentCone, Truncations, Varieties
i1 : load "ChowPolytope.m2";
i2 : R=QQ[s,t,x_0..x_3,MonomialOrder=>Eliminate(2)];
i3 : ElimI = ideal(x_0-s^4,x_1-s^2*t^2,x_2-s*t^3,x_3-t^4);
o3 : Ideal of R
i4 : flatten entries selectInSubring(1, gens gb ElimI)
2 2
o4 = {x - x x , x - x x }
2 1 3 1 0 3
o4 : List
i6 : I = ideal {x_2^2-x_1*x_3, x_1^2-x_0*x_3};
o6 : Ideal of R
i7 : chowPolytope(I)
o7 = {{2, 3, 2, 1}, {3, 0, 4, 1}, {4, 0, 0, 4}, {2, 4, 0, 2}}
o7 : List
i8 : initialIdeals = apply(gfan(I), i -> ideal(first i))
2 3 2
o8 = {ideal (x x , x x , x x ), ideal (x x , x x , x ), ideal (x x , x x ,
1 3 0 3 0 2 0 3 1 3 1 0 3 1 3
---------------------------------------------------------------------
2 3 2 2 2 2 4
x ), ideal (x x , x x , x , x x ), ideal (x x , x , x x , x ), ideal
1 0 3 1 3 1 1 2 1 3 1 1 2 2
---------------------------------------------------------------------
2 2 2
(x x , x ), ideal (x , x )}
0 3 2 2 1
o8 : List
i9 : J=first initialIdeals
2
o9 = ideal (x x , x x , x x )
1 3 0 3 0 2
o9 : Ideal of R
i10 : apply({3,4,5}, m -> complementaryHilbertPoint(m,J))
o10 = {{9, 10, 11, 6}, {16, 19, 19, 10}, {25, 31, 29, 15}}
i11 : outerStatePolynomial(J)
2 3 2 3 2 1 2 1
o11 = {m , -m - -m + 1, m + m - 1, -m + -m}
2 2 2 2
o11 : List
i12 : chowPoint(J)
o12 = {2, 3, 2, 1}
o12 : List
i13 : apply(initialIdeals, J -> chowPoint(J))
o13 = {{2, 3, 2, 1}, {3, 0, 4, 1}, {3, 0, 4, 1}, {3, 0, 4, 1}, {4, 0, 0,
--------------------------------------------------------------------
4}, {2, 4, 0, 2}, {4, 0, 0, 4}}
o13 : List
Approach 3 We compute the secondary polytope. By [KSZ], this is equal to the Chow polytope. Let \(A = \{0,2,3,4\}\), and \(Q = \operatorname{conv}(A) = [0,4]\). The vertices of the secondary polytope \(\Sigma(A)\) correspond to coherent triangulations of \(Q\). Then, for each such triangulation \(T\), we compute the vector \( \phi_T\) whose \(i^{th}\) entry is \[\phi_T(i) = \sum\limits_{\sigma:a_i \in \text{Vert}(\sigma)} \text{Vol}(\sigma) \] In this example, we can enumerate the triangulations and compute these vectors by hand. We obtain the following results. \[ \begin{array}{ll} T & \phi_T \\ \{ [0,4]\} & (4,0,0,4) \\ \{ [0,2],[2,4] \} & (2,4,0,2) \\ \{ [0,3],[3,4]\} & (3,0,4,1) \\ \{ [0,2],[2,3],[3,4]\} & (2,3,2,1) \end{array} \] After computing the Chow polytope, we would compute the proximum and worst 1-PS as we did in Approach 1.
Approach 4 We compute the worst 1-PS using the approach described in our preprint.
Macaulay2, version 1.24.05
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems, Isomorphism, LLLBases,
MinimalPrimes, OnlineLookup, PrimaryDecomposition, ReesAlgebra, Saturation, TangentCone,
Truncations, Varieties
i1 : load "Worst1PS.m2"
i2 : proximumaW(3,{2,3})
88 72 64 8
o2 = {{--, --, --, -}, {}}
35 35 35 5
o2 : List