Examples#
In this section we present some extended examples illustrating various features of the sheaf machinery.
- Example: Big Ex (ex-1d7a31)#
In this example, we consider a surface \(X\) from a special family of rational surfaces of degree 10 in \({\bf P}^4\). This family is described by Decker, Ein and Schreyer in Section 2.1 of [Decker et al., 1993]. They have sectional genus 9 and are isomorphic to the plane blown up in 18 points in special position which give 18 exceptional curves in \(X\). The embedding into \({\bf P}^4\) is such that four of these exceptional curves are of degree 3, seven curves are of degree 2 and seven curves are of degree 1.
The adjunction map on \(X\) is the map corresponding to the divisor \(K_X+H\), where \(K_X\) is a canonical divisor and \(H\) is a hyperplane section, or, equivalently to the sheaf \({\cal K}_X(1)\) where \({\cal K}_X\) is the canonical sheaf. In our example, the adjunction map maps \(X\) to a smooth surface \(X_1\) of degree 13 in \({\bf P}^8\) blowing down the seven degree 1 exceptional curves to points and reducing the degrees of the others by 1. The adjunction map on \(X_1\) blows down the seven exceptional curves originally of degree 2 to points and maps \(X_2\) to an anticanonically embedded degree 5 Del Pezzo surface in \({\bf P}^5\).
We take a randomly generated surface from this family over a small finite field (\({\bf F}_{17}\)) and illustrate this process by explicitly computing the adjunction maps and images \(X_1\) and \(X_2\). We show that the intersection pairings of the canonical divisor and hyperplane sections on \(X\), \(X_1\), \(X_2\) are as expected and that \(X_2\) really is an anticanonically embedded Del Pezzo surface. We also expand the composition of the two divisor maps and show that the resulting map is indeed a birational map from \(X\) onto \(X_2\).
These surfaces \(X\) are defined by one degree 4 and ten degree 5 polynomials in \({\bf P}^4\). The embedding is quite a complex one and it is hard to construct one with defining polynomials which are at all sparse. This makes it fairly challenging for explicit computation and also means that an example takes up a lot of page space! An example with relatively small coefficients over \({\bf Q}\) can also be processed, though the total running time is a few minutes. Also, the resulting \(X_2\) tends to have very large coefficients. Here we get no coefficient blow-up and \(X_2\) is a much simpler looking surface than \(X\).
> P<[x]> := ProjectiveSpace(GF(17),4); > X := Scheme(P, [ > 10*x[1]^4 + 13*x[1]^3*x[2] + 8*x[1]*x[2]^3 + 4*x[2]^4 + 6*x[1]^3*x[3] + > 15*x[1]^2*x[2]*x[3] + 14*x[2]^3*x[3] + x[1]^2*x[3]^2 + > 13*x[1]*x[2]*x[3]^2 + 3*x[2]^2*x[3]^2 + 9*x[1]*x[3]^3 + 2*x[2]*x[3]^3 + > 10*x[3]^4 + 15*x[1]^3*x[4] + 4*x[1]^2*x[2]*x[4] + 3*x[1]*x[2]^2*x[4] + > 7*x[2]^3*x[4] + 9*x[1]^2*x[3]*x[4] + 3*x[1]*x[2]*x[3]*x[4] + > 9*x[2]^2*x[3]*x[4] + 11*x[1]*x[3]^2*x[4] + 6*x[2]*x[3]^2*x[4] + > 15*x[3]^3*x[4] + x[1]^2*x[4]^2 + 4*x[1]*x[2]*x[4]^2 + 2*x[2]^2*x[4]^2 + > 12*x[1]*x[3]*x[4]^2 + 8*x[2]*x[3]*x[4]^2 + 9*x[3]^2*x[4]^2 + > 10*x[1]*x[4]^3 + 5*x[2]*x[4]^3 + 14*x[3]*x[4]^3 + 4*x[1]^3*x[5] + > 16*x[1]^2*x[2]*x[5] + 15*x[2]^3*x[5] + 13*x[1]^2*x[3]*x[5] + > 13*x[1]*x[2]*x[3]*x[5] + 10*x[2]^2*x[3]*x[5] + 15*x[1]*x[3]^2*x[5] + > 7*x[2]*x[3]^2*x[5] + 14*x[3]^3*x[5] + 11*x[1]^2*x[4]*x[5] + > 10*x[1]*x[2]*x[4]*x[5] + 4*x[2]^2*x[4]*x[5] + x[1]*x[3]*x[4]*x[5] + > 12*x[2]*x[3]*x[4]*x[5] + 8*x[3]^2*x[4]*x[5] + 5*x[1]*x[4]^2*x[5] + > 5*x[2]*x[4]^2*x[5] + 11*x[3]*x[4]^2*x[5] + 10*x[4]^3*x[5] + > 12*x[1]^2*x[5]^2 + 8*x[1]*x[2]*x[5]^2 + 16*x[2]^2*x[5]^2 + > 12*x[1]*x[3]*x[5]^2 + x[2]*x[3]*x[5]^2 + 14*x[3]^2*x[5]^2 + > 8*x[1]*x[4]*x[5]^2 + x[2]*x[4]*x[5]^2 + 3*x[3]*x[4]*x[5]^2 + > 5*x[4]^2*x[5]^2 + 11*x[1]*x[5]^3 + 13*x[2]*x[5]^3 + 5*x[3]*x[5]^3 + > 9*x[4]*x[5]^3 + 8*x[5]^4, > 9*x[1]^4*x[4] + 14*x[1]^3*x[2]*x[4] + 5*x[1]^2*x[2]^2*x[4] + > 2*x[1]*x[2]^3*x[4] + 2*x[1]^3*x[3]*x[4] + 7*x[1]^2*x[2]*x[3]*x[4] + > 5*x[1]*x[2]^2*x[3]*x[4] + 7*x[2]^3*x[3]*x[4] + 9*x[1]^2*x[3]^2*x[4] + > 12*x[1]*x[2]*x[3]^2*x[4] + 2*x[2]^2*x[3]^2*x[4] + 9*x[1]*x[3]^3*x[4] + > 2*x[2]*x[3]^3*x[4] + x[3]^4*x[4] + 3*x[1]^3*x[4]^2 + > 5*x[1]^2*x[2]*x[4]^2 + 7*x[1]*x[2]^2*x[4]^2 + 13*x[2]^3*x[4]^2 + > 11*x[1]^2*x[3]*x[4]^2 + 4*x[1]*x[2]*x[3]*x[4]^2 + 11*x[2]^2*x[3]*x[4]^2 > + 14*x[1]*x[3]^2*x[4]^2 + 16*x[2]*x[3]^2*x[4]^2 + 15*x[1]^2*x[4]^3 + > 11*x[1]*x[2]*x[4]^3 + 5*x[2]^2*x[4]^3 + 6*x[1]*x[3]*x[4]^3 + > 9*x[2]*x[3]*x[4]^3 + 16*x[3]^2*x[4]^3 + 9*x[2]*x[4]^4 + 15*x[3]*x[4]^4 + > 14*x[4]^5 + 2*x[1]^3*x[2]*x[5] + 6*x[1]^2*x[2]^2*x[5] + > 3*x[1]*x[2]^3*x[5] + 16*x[2]^4*x[5] + 15*x[1]^3*x[3]*x[5] + > 6*x[1]*x[2]^2*x[3]*x[5] + 10*x[2]^3*x[3]*x[5] + 14*x[1]^2*x[3]^2*x[5] + > 13*x[1]*x[2]*x[3]^2*x[5] + 4*x[2]^2*x[3]^2*x[5] + 16*x[1]*x[3]^3*x[5] + > 13*x[3]^4*x[5] + 14*x[1]^3*x[4]*x[5] + 9*x[1]^2*x[2]*x[4]*x[5] + > 16*x[1]*x[2]^2*x[4]*x[5] + 14*x[2]^3*x[4]*x[5] + > 6*x[1]*x[2]*x[3]*x[4]*x[5] + 6*x[2]^2*x[3]*x[4]*x[5] + > 3*x[1]*x[3]^2*x[4]*x[5] + 7*x[2]*x[3]^2*x[4]*x[5] + 7*x[3]^3*x[4]*x[5] + > 2*x[1]^2*x[4]^2*x[5] + 15*x[1]*x[2]*x[4]^2*x[5] + > 9*x[1]*x[3]*x[4]^2*x[5] + 14*x[3]^2*x[4]^2*x[5] + 14*x[1]*x[4]^3*x[5] + > 6*x[2]*x[4]^3*x[5] + 12*x[3]*x[4]^3*x[5] + 3*x[4]^4*x[5] + > 9*x[1]^3*x[5]^2 + 12*x[1]^2*x[2]*x[5]^2 + 16*x[1]*x[2]^2*x[5]^2 + > x[2]^3*x[5]^2 + 7*x[1]^2*x[3]*x[5]^2 + 5*x[1]*x[2]*x[3]*x[5]^2 + > 8*x[2]^2*x[3]*x[5]^2 + 2*x[1]*x[3]^2*x[5]^2 + 4*x[2]*x[3]^2*x[5]^2 + > 13*x[3]^3*x[5]^2 + 7*x[1]^2*x[4]*x[5]^2 + 6*x[2]^2*x[4]*x[5]^2 + > 16*x[1]*x[3]*x[4]*x[5]^2 + 15*x[2]*x[3]*x[4]*x[5]^2 + > 7*x[3]^2*x[4]*x[5]^2 + 6*x[1]*x[4]^2*x[5]^2 + 3*x[2]*x[4]^2*x[5]^2 + > 16*x[3]*x[4]^2*x[5]^2 + 15*x[4]^3*x[5]^2 + x[1]^2*x[5]^3 + > 13*x[1]*x[2]*x[5]^3 + 6*x[2]^2*x[5]^3 + 8*x[1]*x[3]*x[5]^3 + > x[2]*x[3]*x[5]^3 + 9*x[3]^2*x[5]^3 + 3*x[1]*x[4]*x[5]^3 + > 14*x[2]*x[4]*x[5]^3 + 8*x[3]*x[4]*x[5]^3 + 14*x[4]^2*x[5]^3 + > 16*x[1]*x[5]^4 + 2*x[2]*x[5]^4 + 7*x[3]*x[5]^4 + 7*x[4]*x[5]^4 + > 11*x[5]^5, > 13*x[1]^4*x[4] + 8*x[1]^3*x[2]*x[4] + 14*x[1]^2*x[2]^2*x[4] + > 3*x[1]*x[2]^3*x[4] + 11*x[2]^4*x[4] + 7*x[1]^3*x[3]*x[4] + > 3*x[1]^2*x[2]*x[3]*x[4] + 12*x[2]^3*x[3]*x[4] + 3*x[1]^2*x[3]^2*x[4] + > 13*x[1]*x[2]*x[3]^2*x[4] + 3*x[2]^2*x[3]^2*x[4] + 7*x[1]*x[3]^3*x[4] + > 2*x[2]*x[3]^3*x[4] + 7*x[3]^4*x[4] + 13*x[1]^3*x[4]^2 + > 6*x[1]^2*x[2]*x[4]^2 + 6*x[1]*x[2]^2*x[4]^2 + 6*x[2]^3*x[4]^2 + > 2*x[1]^2*x[3]*x[4]^2 + 15*x[1]*x[2]*x[3]*x[4]^2 + 14*x[2]^2*x[3]*x[4]^2 > + 3*x[1]*x[3]^2*x[4]^2 + 16*x[2]*x[3]^2*x[4]^2 + 3*x[3]^3*x[4]^2 + > 6*x[1]^2*x[4]^3 + 10*x[2]^2*x[4]^3 + 7*x[2]*x[3]*x[4]^3 + 13*x[1]*x[4]^4 > + 5*x[2]*x[4]^4 + 15*x[3]*x[4]^4 + 13*x[4]^5 + 2*x[1]^4*x[5] + > 6*x[1]^3*x[2]*x[5] + 12*x[1]^2*x[2]^2*x[5] + 12*x[1]*x[2]^3*x[5] + > 2*x[2]^4*x[5] + 5*x[1]^3*x[3]*x[5] + 12*x[1]^2*x[2]*x[3]*x[5] + > 7*x[1]*x[2]^2*x[3]*x[5] + 11*x[2]^3*x[3]*x[5] + 2*x[1]^2*x[3]^2*x[5] + > 3*x[1]*x[2]*x[3]^2*x[5] + 7*x[2]^2*x[3]^2*x[5] + 16*x[1]*x[3]^3*x[5] + > 3*x[2]*x[3]^3*x[5] + 13*x[3]^4*x[5] + 2*x[1]^2*x[2]*x[4]*x[5] + > 12*x[1]*x[2]^2*x[4]*x[5] + 2*x[2]^3*x[4]*x[5] + 10*x[1]^2*x[3]*x[4]*x[5] > + 9*x[1]*x[2]*x[3]*x[4]*x[5] + 6*x[2]^2*x[3]*x[4]*x[5] + > x[1]*x[3]^2*x[4]*x[5] + 6*x[2]*x[3]^2*x[4]*x[5] + 15*x[3]^3*x[4]*x[5] + > 2*x[1]^2*x[4]^2*x[5] + 14*x[1]*x[2]*x[4]^2*x[5] + > 13*x[1]*x[3]*x[4]^2*x[5] + 13*x[2]*x[3]*x[4]^2*x[5] + > 2*x[3]^2*x[4]^2*x[5] + 12*x[1]*x[4]^3*x[5] + 8*x[2]*x[4]^3*x[5] + > 8*x[3]*x[4]^3*x[5] + x[4]^4*x[5] + 3*x[1]^3*x[5]^2 + > 7*x[1]^2*x[2]*x[5]^2 + 4*x[1]^2*x[3]*x[5]^2 + 3*x[1]*x[2]*x[3]*x[5]^2 + > 9*x[2]^2*x[3]*x[5]^2 + 14*x[1]*x[3]^2*x[5]^2 + 13*x[2]*x[3]^2*x[5]^2 + > 15*x[3]^3*x[5]^2 + x[1]^2*x[4]*x[5]^2 + 14*x[1]*x[2]*x[4]*x[5]^2 + > 5*x[2]^2*x[4]*x[5]^2 + 10*x[1]*x[3]*x[4]*x[5]^2 + > 5*x[2]*x[3]*x[4]*x[5]^2 + 7*x[3]^2*x[4]*x[5]^2 + 13*x[1]*x[4]^2*x[5]^2 + > 2*x[2]*x[4]^2*x[5]^2 + 9*x[3]*x[4]^2*x[5]^2 + 3*x[4]^3*x[5]^2 + > 14*x[1]*x[2]*x[5]^3 + 12*x[2]^2*x[5]^3 + 6*x[1]*x[3]*x[5]^3 + > 16*x[2]*x[3]*x[5]^3 + 8*x[3]^2*x[5]^3 + 3*x[1]*x[4]*x[5]^3 + > 4*x[2]*x[4]*x[5]^3 + 11*x[3]*x[4]*x[5]^3 + 15*x[4]^2*x[5]^3 + > 14*x[1]*x[5]^4 + 13*x[2]*x[5]^4 + 4*x[3]*x[5]^4 + 4*x[4]*x[5]^4 + > 13*x[5]^5, > 15*x[1]^3*x[2]*x[3] + 11*x[1]^2*x[2]^2*x[3] + 14*x[1]*x[2]^3*x[3] + > x[2]^4*x[3] + 2*x[1]^3*x[3]^2 + 11*x[1]*x[2]^2*x[3]^2 + 7*x[2]^3*x[3]^2 > + 3*x[1]^2*x[3]^3 + 4*x[1]*x[2]*x[3]^3 + 13*x[2]^2*x[3]^3 + x[1]*x[3]^4 > + 4*x[3]^5 + 2*x[1]^4*x[4] + 11*x[1]^3*x[2]*x[4] + 13*x[1]^2*x[2]^2*x[4] > + 4*x[1]*x[2]^3*x[4] + 16*x[2]^4*x[4] + 5*x[1]^3*x[3]*x[4] + > 4*x[1]^2*x[2]*x[3]*x[4] + 10*x[1]*x[2]^2*x[3]*x[4] + 8*x[2]^3*x[3]*x[4] > + 5*x[1]^2*x[3]^2*x[4] + 14*x[1]*x[2]*x[3]^2*x[4] + 2*x[2]^2*x[3]^2*x[4] > + 15*x[1]*x[3]^3*x[4] + 13*x[3]^4*x[4] + 9*x[1]^3*x[4]^2 + > 3*x[1]^2*x[2]*x[4]^2 + 10*x[1]*x[2]^2*x[4]^2 + 12*x[2]^3*x[4]^2 + > 8*x[1]^2*x[3]*x[4]^2 + 14*x[1]*x[2]*x[3]*x[4]^2 + 3*x[2]^2*x[3]*x[4]^2 + > 2*x[1]*x[3]^2*x[4]^2 + 5*x[2]*x[3]^2*x[4]^2 + 10*x[3]^3*x[4]^2 + > 5*x[1]^2*x[4]^3 + x[1]*x[2]*x[4]^3 + 8*x[2]^2*x[4]^3 + > 7*x[1]*x[3]*x[4]^3 + 10*x[2]*x[3]*x[4]^3 + 13*x[3]^2*x[4]^3 + > 10*x[1]*x[4]^4 + 7*x[2]*x[4]^4 + 16*x[3]*x[4]^4 + 16*x[4]^5 + > 8*x[1]^3*x[3]*x[5] + 5*x[1]^2*x[2]*x[3]*x[5] + x[1]*x[2]^2*x[3]*x[5] + > 16*x[2]^3*x[3]*x[5] + 10*x[1]^2*x[3]^2*x[5] + 12*x[1]*x[2]*x[3]^2*x[5] + > 9*x[2]^2*x[3]^2*x[5] + 15*x[1]*x[3]^3*x[5] + 13*x[2]*x[3]^3*x[5] + > 4*x[3]^4*x[5] + 14*x[1]^3*x[4]*x[5] + x[1]^2*x[2]*x[4]*x[5] + > 10*x[1]*x[2]^2*x[4]*x[5] + 11*x[2]^3*x[4]*x[5] + 5*x[1]^2*x[3]*x[4]*x[5] > + 12*x[1]*x[2]*x[3]*x[4]*x[5] + 7*x[2]^2*x[3]*x[4]*x[5] + > 5*x[1]*x[3]^2*x[4]*x[5] + 3*x[3]^3*x[4]*x[5] + 2*x[1]^2*x[4]^2*x[5] + > 5*x[2]^2*x[4]^2*x[5] + 2*x[2]*x[3]*x[4]^2*x[5] + 8*x[3]^2*x[4]^2*x[5] + > x[1]*x[4]^3*x[5] + 5*x[2]*x[4]^3*x[5] + 3*x[3]*x[4]^3*x[5] + > 14*x[4]^4*x[5] + 16*x[1]^2*x[3]*x[5]^2 + 4*x[1]*x[2]*x[3]*x[5]^2 + > 11*x[2]^2*x[3]*x[5]^2 + 9*x[1]*x[3]^2*x[5]^2 + 16*x[2]*x[3]^2*x[5]^2 + > 8*x[3]^3*x[5]^2 + 8*x[1]^2*x[4]*x[5]^2 + 11*x[1]*x[2]*x[4]*x[5]^2 + > 3*x[2]^2*x[4]*x[5]^2 + 6*x[1]*x[3]*x[4]*x[5]^2 + 9*x[2]*x[3]*x[4]*x[5]^2 > + 5*x[3]^2*x[4]*x[5]^2 + 15*x[1]*x[4]^2*x[5]^2 + 2*x[2]*x[4]^2*x[5]^2 + > 8*x[3]*x[4]^2*x[5]^2 + 14*x[4]^3*x[5]^2 + x[1]*x[3]*x[5]^3 + > 15*x[2]*x[3]*x[5]^3 + 10*x[3]^2*x[5]^3 + 11*x[1]*x[4]*x[5]^3 + > 8*x[2]*x[4]*x[5]^3 + 15*x[3]*x[4]*x[5]^3 + 15*x[4]^2*x[5]^3 + > 6*x[3]*x[5]^4 + 3*x[4]*x[5]^4, > 9*x[1]^4*x[3] + 14*x[1]^3*x[2]*x[3] + 5*x[1]^2*x[2]^2*x[3] + > 2*x[1]*x[2]^3*x[3] + 2*x[1]^3*x[3]^2 + 7*x[1]^2*x[2]*x[3]^2 + > 5*x[1]*x[2]^2*x[3]^2 + 7*x[2]^3*x[3]^2 + 9*x[1]^2*x[3]^3 + > 12*x[1]*x[2]*x[3]^3 + 2*x[2]^2*x[3]^3 + 9*x[1]*x[3]^4 + 2*x[2]*x[3]^4 + > x[3]^5 + 3*x[1]^3*x[3]*x[4] + 5*x[1]^2*x[2]*x[3]*x[4] + > 7*x[1]*x[2]^2*x[3]*x[4] + 13*x[2]^3*x[3]*x[4] + 11*x[1]^2*x[3]^2*x[4] + > 4*x[1]*x[2]*x[3]^2*x[4] + 11*x[2]^2*x[3]^2*x[4] + 14*x[1]*x[3]^3*x[4] + > 16*x[2]*x[3]^3*x[4] + 15*x[1]^2*x[3]*x[4]^2 + 11*x[1]*x[2]*x[3]*x[4]^2 + > 5*x[2]^2*x[3]*x[4]^2 + 6*x[1]*x[3]^2*x[4]^2 + 9*x[2]*x[3]^2*x[4]^2 + > 16*x[3]^3*x[4]^2 + 9*x[2]*x[3]*x[4]^3 + 15*x[3]^2*x[4]^3 + > 14*x[3]*x[4]^4 + 2*x[1]^4*x[5] + 11*x[1]^3*x[2]*x[5] + > 13*x[1]^2*x[2]^2*x[5] + 4*x[1]*x[2]^3*x[5] + 16*x[2]^4*x[5] + > 2*x[1]^3*x[3]*x[5] + 13*x[1]^2*x[2]*x[3]*x[5] + 9*x[1]*x[2]^2*x[3]*x[5] > + 5*x[2]^3*x[3]*x[5] + 5*x[1]^2*x[3]^2*x[5] + 3*x[1]*x[2]*x[3]^2*x[5] + > 8*x[2]^2*x[3]^2*x[5] + x[1]*x[3]^3*x[5] + 7*x[2]*x[3]^3*x[5] + > 3*x[3]^4*x[5] + 9*x[1]^3*x[4]*x[5] + 3*x[1]^2*x[2]*x[4]*x[5] + > 10*x[1]*x[2]^2*x[4]*x[5] + 12*x[2]^3*x[4]*x[5] + > 10*x[1]^2*x[3]*x[4]*x[5] + 12*x[1]*x[2]*x[3]*x[4]*x[5] + > 3*x[2]^2*x[3]*x[4]*x[5] + 11*x[1]*x[3]^2*x[4]*x[5] + > 5*x[2]*x[3]^2*x[4]*x[5] + 7*x[3]^3*x[4]*x[5] + 5*x[1]^2*x[4]^2*x[5] + > x[1]*x[2]*x[4]^2*x[5] + 8*x[2]^2*x[4]^2*x[5] + 4*x[1]*x[3]*x[4]^2*x[5] + > 16*x[2]*x[3]*x[4]^2*x[5] + 8*x[3]^2*x[4]^2*x[5] + 10*x[1]*x[4]^3*x[5] + > 7*x[2]*x[4]^3*x[5] + 2*x[3]*x[4]^3*x[5] + 16*x[4]^4*x[5] + > 14*x[1]^3*x[5]^2 + x[1]^2*x[2]*x[5]^2 + 10*x[1]*x[2]^2*x[5]^2 + > 11*x[2]^3*x[5]^2 + 12*x[1]^2*x[3]*x[5]^2 + 12*x[1]*x[2]*x[3]*x[5]^2 + > 13*x[2]^2*x[3]*x[5]^2 + 4*x[1]*x[3]^2*x[5]^2 + 15*x[2]*x[3]^2*x[5]^2 + > 10*x[3]^3*x[5]^2 + 2*x[1]^2*x[4]*x[5]^2 + 5*x[2]^2*x[4]*x[5]^2 + > 6*x[1]*x[3]*x[4]*x[5]^2 + 5*x[2]*x[3]*x[4]*x[5]^2 + 7*x[3]^2*x[4]*x[5]^2 > + x[1]*x[4]^2*x[5]^2 + 5*x[2]*x[4]^2*x[5]^2 + x[3]*x[4]^2*x[5]^2 + > 14*x[4]^3*x[5]^2 + 8*x[1]^2*x[5]^3 + 11*x[1]*x[2]*x[5]^3 + > 3*x[2]^2*x[5]^3 + 9*x[1]*x[3]*x[5]^3 + 6*x[2]*x[3]*x[5]^3 + > 13*x[3]^2*x[5]^3 + 15*x[1]*x[4]*x[5]^3 + 2*x[2]*x[4]*x[5]^3 + > 5*x[3]*x[4]*x[5]^3 + 14*x[4]^2*x[5]^3 + 11*x[1]*x[5]^4 + 8*x[2]*x[5]^4 + > 5*x[3]*x[5]^4 + 15*x[4]*x[5]^4 + 3*x[5]^5, > 13*x[1]^4*x[3] + 8*x[1]^3*x[2]*x[3] + 14*x[1]^2*x[2]^2*x[3] + > 3*x[1]*x[2]^3*x[3] + 11*x[2]^4*x[3] + 7*x[1]^3*x[3]^2 + > 3*x[1]^2*x[2]*x[3]^2 + 12*x[2]^3*x[3]^2 + 3*x[1]^2*x[3]^3 + > 13*x[1]*x[2]*x[3]^3 + 3*x[2]^2*x[3]^3 + 7*x[1]*x[3]^4 + 2*x[2]*x[3]^4 + > 7*x[3]^5 + 13*x[1]^3*x[3]*x[4] + 6*x[1]^2*x[2]*x[3]*x[4] + > 6*x[1]*x[2]^2*x[3]*x[4] + 6*x[2]^3*x[3]*x[4] + 2*x[1]^2*x[3]^2*x[4] + > 15*x[1]*x[2]*x[3]^2*x[4] + 14*x[2]^2*x[3]^2*x[4] + 3*x[1]*x[3]^3*x[4] + > 16*x[2]*x[3]^3*x[4] + 3*x[3]^4*x[4] + 6*x[1]^2*x[3]*x[4]^2 + > 10*x[2]^2*x[3]*x[4]^2 + 7*x[2]*x[3]^2*x[4]^2 + 13*x[1]*x[3]*x[4]^3 + > 5*x[2]*x[3]*x[4]^3 + 15*x[3]^2*x[4]^3 + 13*x[3]*x[4]^4 + 15*x[1]^4*x[5] > + 15*x[1]^3*x[2]*x[5] + 2*x[1]^2*x[2]^2*x[5] + 16*x[1]*x[2]^3*x[5] + > 16*x[2]^4*x[5] + 14*x[1]^3*x[3]*x[5] + 4*x[1]^2*x[2]*x[3]*x[5] + > 10*x[1]*x[2]^2*x[3]*x[5] + 4*x[2]^3*x[3]*x[5] + 8*x[1]^2*x[3]^2*x[5] + > 5*x[1]*x[2]*x[3]^2*x[5] + 11*x[2]^2*x[3]^2*x[5] + 12*x[2]*x[3]^3*x[5] + > 2*x[3]^4*x[5] + 15*x[1]^2*x[2]*x[4]*x[5] + 6*x[2]^3*x[4]*x[5] + > 9*x[1]^2*x[3]*x[4]*x[5] + 9*x[1]*x[2]*x[3]*x[4]*x[5] + > 15*x[2]^2*x[3]*x[4]*x[5] + 14*x[1]*x[3]^2*x[4]*x[5] + > 13*x[2]*x[3]^2*x[4]*x[5] + 6*x[3]^3*x[4]*x[5] + 4*x[1]^2*x[4]^2*x[5] + > 7*x[1]*x[2]*x[4]^2*x[5] + 3*x[2]^2*x[4]^2*x[5] + 8*x[1]*x[3]*x[4]^2*x[5] > + 8*x[2]*x[3]*x[4]^2*x[5] + 3*x[3]^2*x[4]^2*x[5] + 15*x[1]*x[4]^3*x[5] + > 3*x[2]*x[4]^3*x[5] + 8*x[3]*x[4]^3*x[5] + 2*x[4]^4*x[5] + > 2*x[1]^3*x[5]^2 + 6*x[1]^2*x[2]*x[5]^2 + x[1]*x[2]^2*x[5]^2 + > 7*x[2]^3*x[5]^2 + 3*x[1]^2*x[3]*x[5]^2 + 16*x[1]*x[2]*x[3]*x[5]^2 + > 10*x[2]^2*x[3]*x[5]^2 + 10*x[1]*x[3]^2*x[5]^2 + 13*x[2]*x[3]^2*x[5]^2 + > 2*x[3]^3*x[5]^2 + 4*x[1]^2*x[4]*x[5]^2 + x[1]*x[2]*x[4]*x[5]^2 + > 9*x[2]^2*x[4]*x[5]^2 + 16*x[1]*x[3]*x[4]*x[5]^2 + > 8*x[2]*x[3]*x[4]*x[5]^2 + 11*x[1]*x[4]^2*x[5]^2 + 11*x[2]*x[4]^2*x[5]^2 > + 4*x[3]*x[4]^2*x[5]^2 + 10*x[4]^3*x[5]^2 + 10*x[1]^2*x[5]^3 + > 14*x[2]^2*x[5]^3 + 16*x[1]*x[3]*x[5]^3 + 13*x[2]*x[3]*x[5]^3 + > 15*x[3]^2*x[5]^3 + 16*x[1]*x[4]*x[5]^3 + 3*x[2]*x[4]*x[5]^3 + > 4*x[3]*x[4]*x[5]^3 + 2*x[4]^2*x[5]^3 + x[1]*x[5]^4 + 7*x[2]*x[5]^4 + > 7*x[4]*x[5]^4 + 2*x[5]^5, > 15*x[1]^4*x[3] + 11*x[1]^3*x[2]*x[3] + 5*x[1]^2*x[2]^2*x[3] + > 5*x[1]*x[2]^3*x[3] + 15*x[2]^4*x[3] + 12*x[1]^3*x[3]^2 + > 5*x[1]^2*x[2]*x[3]^2 + 10*x[1]*x[2]^2*x[3]^2 + 6*x[2]^3*x[3]^2 + > 15*x[1]^2*x[3]^3 + 14*x[1]*x[2]*x[3]^3 + 10*x[2]^2*x[3]^3 + x[1]*x[3]^4 > + 14*x[2]*x[3]^4 + 4*x[3]^5 + 15*x[1]^4*x[4] + 15*x[1]^3*x[2]*x[4] + > 2*x[1]^2*x[2]^2*x[4] + 16*x[1]*x[2]^3*x[4] + 16*x[2]^4*x[4] + > 14*x[1]^3*x[3]*x[4] + 2*x[1]^2*x[2]*x[3]*x[4] + 15*x[1]*x[2]^2*x[3]*x[4] > + 2*x[2]^3*x[3]*x[4] + 15*x[1]^2*x[3]^2*x[4] + 13*x[1]*x[2]*x[3]^2*x[4] > + 5*x[2]^2*x[3]^2*x[4] + 16*x[1]*x[3]^3*x[4] + 6*x[2]*x[3]^3*x[4] + > 4*x[3]^4*x[4] + 15*x[1]^2*x[2]*x[4]^2 + 6*x[2]^3*x[4]^2 + > 7*x[1]^2*x[3]*x[4]^2 + 12*x[1]*x[2]*x[3]*x[4]^2 + 15*x[2]^2*x[3]*x[4]^2 > + x[1]*x[3]^2*x[4]^2 + 4*x[3]^3*x[4]^2 + 4*x[1]^2*x[4]^3 + > 7*x[1]*x[2]*x[4]^3 + 3*x[2]^2*x[4]^3 + 13*x[1]*x[3]*x[4]^3 + > 12*x[3]^2*x[4]^3 + 15*x[1]*x[4]^4 + 3*x[2]*x[4]^4 + 7*x[3]*x[4]^4 + > 2*x[4]^5 + 14*x[1]^3*x[3]*x[5] + 10*x[1]^2*x[2]*x[3]*x[5] + > 13*x[1]^2*x[3]^2*x[5] + 14*x[1]*x[2]*x[3]^2*x[5] + 8*x[2]^2*x[3]^2*x[5] > + 3*x[1]*x[3]^3*x[5] + 4*x[2]*x[3]^3*x[5] + 2*x[3]^4*x[5] + > 2*x[1]^3*x[4]*x[5] + 6*x[1]^2*x[2]*x[4]*x[5] + x[1]*x[2]^2*x[4]*x[5] + > 7*x[2]^3*x[4]*x[5] + 2*x[1]^2*x[3]*x[4]*x[5] + > 2*x[1]*x[2]*x[3]*x[4]*x[5] + 5*x[2]^2*x[3]*x[4]*x[5] + > 8*x[2]*x[3]^2*x[4]*x[5] + 12*x[3]^3*x[4]*x[5] + 4*x[1]^2*x[4]^2*x[5] + > x[1]*x[2]*x[4]^2*x[5] + 9*x[2]^2*x[4]^2*x[5] + 3*x[1]*x[3]*x[4]^2*x[5] + > 6*x[2]*x[3]*x[4]^2*x[5] + 8*x[3]^2*x[4]^2*x[5] + 11*x[1]*x[4]^3*x[5] + > 11*x[2]*x[4]^3*x[5] + x[3]*x[4]^3*x[5] + 10*x[4]^4*x[5] + > 3*x[1]*x[2]*x[3]*x[5]^2 + 5*x[2]^2*x[3]*x[5]^2 + 11*x[1]*x[3]^2*x[5]^2 + > x[2]*x[3]^2*x[5]^2 + 9*x[3]^3*x[5]^2 + 10*x[1]^2*x[4]*x[5]^2 + > 14*x[2]^2*x[4]*x[5]^2 + 13*x[1]*x[3]*x[4]*x[5]^2 + > 9*x[2]*x[3]*x[4]*x[5]^2 + 4*x[3]^2*x[4]*x[5]^2 + 16*x[1]*x[4]^2*x[5]^2 + > 3*x[2]*x[4]^2*x[5]^2 + 6*x[3]*x[4]^2*x[5]^2 + 2*x[4]^3*x[5]^2 + > 3*x[1]*x[3]*x[5]^3 + 4*x[2]*x[3]*x[5]^3 + 13*x[3]^2*x[5]^3 + > x[1]*x[4]*x[5]^3 + 7*x[2]*x[4]*x[5]^3 + 13*x[3]*x[4]*x[5]^3 + > 7*x[4]^2*x[5]^3 + 4*x[3]*x[5]^4 + 2*x[4]*x[5]^4, > 15*x[1]^3*x[2]^2 + 11*x[1]^2*x[2]^3 + 14*x[1]*x[2]^4 + x[2]^5 + > 2*x[1]^3*x[2]*x[3] + 11*x[1]*x[2]^3*x[3] + 7*x[2]^4*x[3] + > 3*x[1]^2*x[2]*x[3]^2 + 4*x[1]*x[2]^2*x[3]^2 + 13*x[2]^3*x[3]^2 + > x[1]*x[2]*x[3]^3 + 4*x[2]*x[3]^4 + 16*x[1]^4*x[4] + 2*x[1]^3*x[2]*x[4] + > 7*x[1]^2*x[2]^2*x[4] + 4*x[1]*x[2]^3*x[4] + 4*x[2]^4*x[4] + > 9*x[1]^3*x[3]*x[4] + x[1]^2*x[2]*x[3]*x[4] + 9*x[1]*x[2]^2*x[3]*x[4] + > 4*x[2]^3*x[3]*x[4] + 7*x[1]^2*x[3]^2*x[4] + 5*x[1]*x[2]*x[3]^2*x[4] + > 11*x[2]^2*x[3]^2*x[4] + 15*x[1]*x[3]^3*x[4] + 15*x[2]*x[3]^3*x[4] + > x[3]^4*x[4] + 9*x[1]^2*x[2]*x[4]^2 + 16*x[1]*x[2]^2*x[4]^2 + > 9*x[2]^3*x[4]^2 + 3*x[1]^2*x[3]*x[4]^2 + 2*x[1]*x[2]*x[3]*x[4]^2 + > 14*x[2]^2*x[3]*x[4]^2 + 11*x[1]*x[3]^2*x[4]^2 + 16*x[2]*x[3]^2*x[4]^2 + > 4*x[3]^3*x[4]^2 + x[1]^2*x[4]^3 + 8*x[1]*x[2]*x[4]^3 + 14*x[2]^2*x[4]^3 > + 3*x[1]*x[3]*x[4]^3 + 16*x[2]*x[3]*x[4]^3 + 12*x[3]^2*x[4]^3 + > 7*x[1]*x[4]^4 + 5*x[2]*x[4]^4 + 4*x[3]*x[4]^4 + 2*x[4]^5 + > 8*x[1]^3*x[2]*x[5] + 5*x[1]^2*x[2]^2*x[5] + x[1]*x[2]^3*x[5] + > 16*x[2]^4*x[5] + 10*x[1]^2*x[2]*x[3]*x[5] + 12*x[1]*x[2]^2*x[3]*x[5] + > 9*x[2]^3*x[3]*x[5] + 15*x[1]*x[2]*x[3]^2*x[5] + 13*x[2]^2*x[3]^2*x[5] + > 4*x[2]*x[3]^3*x[5] + 8*x[1]^3*x[4]*x[5] + 16*x[1]^2*x[2]*x[4]*x[5] + > 11*x[1]*x[2]^2*x[4]*x[5] + 8*x[2]^3*x[4]*x[5] + 10*x[1]^2*x[3]*x[4]*x[5] > + 15*x[1]*x[2]*x[3]*x[4]*x[5] + 4*x[2]^2*x[3]*x[4]*x[5] + > 9*x[1]*x[3]^2*x[4]*x[5] + 16*x[2]*x[3]^2*x[4]*x[5] + 11*x[3]^3*x[4]*x[5] > + 4*x[1]^2*x[4]^2*x[5] + 6*x[1]*x[2]*x[4]^2*x[5] + 10*x[2]^2*x[4]^2*x[5] > + 11*x[1]*x[3]*x[4]^2*x[5] + 11*x[2]*x[3]*x[4]^2*x[5] + > 14*x[3]^2*x[4]^2*x[5] + 10*x[1]*x[4]^3*x[5] + 6*x[2]*x[4]^3*x[5] + > 5*x[3]*x[4]^3*x[5] + 4*x[4]^4*x[5] + 16*x[1]^2*x[2]*x[5]^2 + > 4*x[1]*x[2]^2*x[5]^2 + 11*x[2]^3*x[5]^2 + 9*x[1]*x[2]*x[3]*x[5]^2 + > 16*x[2]^2*x[3]*x[5]^2 + 8*x[2]*x[3]^2*x[5]^2 + 3*x[1]^2*x[4]*x[5]^2 + > 10*x[1]*x[2]*x[4]*x[5]^2 + 9*x[2]^2*x[4]*x[5]^2 + > 10*x[1]*x[3]*x[4]*x[5]^2 + 11*x[2]*x[3]*x[4]*x[5]^2 + > 11*x[3]^2*x[4]*x[5]^2 + 3*x[1]*x[4]^2*x[5]^2 + 14*x[2]*x[4]^2*x[5]^2 + > 7*x[3]*x[4]^2*x[5]^2 + 3*x[4]^3*x[5]^2 + x[1]*x[2]*x[5]^3 + > 15*x[2]^2*x[5]^3 + 10*x[2]*x[3]*x[5]^3 + x[1]*x[4]*x[5]^3 + > 2*x[2]*x[4]*x[5]^3 + 5*x[3]*x[4]*x[5]^3 + 6*x[4]^2*x[5]^3 + > 6*x[2]*x[5]^4 + 16*x[4]*x[5]^4, > 2*x[1]^4*x[2] + 2*x[1]^3*x[2]^2 + 15*x[1]^2*x[2]^3 + x[1]*x[2]^4 + x[2]^5 + > 16*x[1]^4*x[3] + 8*x[1]^3*x[2]*x[3] + 7*x[1]^2*x[2]^2*x[3] + > 10*x[1]*x[2]^3*x[3] + 10*x[2]^4*x[3] + 10*x[1]^3*x[3]^2 + > 11*x[1]^2*x[2]*x[3]^2 + 9*x[1]*x[2]^2*x[3]^2 + 10*x[2]^3*x[3]^2 + > 7*x[1]^2*x[3]^3 + 10*x[1]*x[2]*x[3]^3 + 4*x[2]^2*x[3]^3 + 13*x[1]*x[3]^4 > + 3*x[3]^5 + 2*x[1]^2*x[2]^2*x[4] + 11*x[2]^4*x[4] + 5*x[1]^3*x[3]*x[4] > + 15*x[1]^2*x[2]*x[3]*x[4] + 16*x[1]*x[2]^2*x[3]*x[4] + > 4*x[2]^3*x[3]*x[4] + 16*x[1]^2*x[3]^2*x[4] + 12*x[1]*x[2]*x[3]^2*x[4] + > 4*x[2]^2*x[3]^2*x[4] + 15*x[1]*x[3]^3*x[4] + 14*x[2]*x[3]^3*x[4] + > 5*x[3]^4*x[4] + 13*x[1]^2*x[2]*x[4]^2 + 10*x[1]*x[2]^2*x[4]^2 + > 14*x[2]^3*x[4]^2 + 10*x[1]^2*x[3]*x[4]^2 + 9*x[1]*x[2]*x[3]*x[4]^2 + > 2*x[2]^2*x[3]*x[4]^2 + x[1]*x[3]^2*x[4]^2 + 15*x[2]*x[3]^2*x[4]^2 + > 15*x[3]^3*x[4]^2 + 2*x[1]*x[2]*x[4]^3 + 14*x[2]^2*x[4]^3 + > 15*x[1]*x[3]*x[4]^3 + 6*x[2]*x[3]*x[4]^3 + 3*x[3]^2*x[4]^3 + > 15*x[2]*x[4]^4 + 13*x[3]*x[4]^4 + 15*x[1]^3*x[2]*x[5] + > 11*x[1]^2*x[2]^2*x[5] + 16*x[1]*x[2]^3*x[5] + 10*x[2]^4*x[5] + > 4*x[1]^3*x[3]*x[5] + 10*x[1]^2*x[2]*x[3]*x[5] + 4*x[2]^3*x[3]*x[5] + > 3*x[1]^2*x[3]^2*x[5] + 6*x[1]*x[2]*x[3]^2*x[5] + 15*x[2]^2*x[3]^2*x[5] + > 8*x[1]*x[3]^3*x[5] + 8*x[2]*x[3]^3*x[5] + x[3]^4*x[5] + > 13*x[1]^2*x[2]*x[4]*x[5] + 16*x[1]*x[2]^2*x[4]*x[5] + 8*x[2]^3*x[4]*x[5] > + 5*x[1]^2*x[3]*x[4]*x[5] + 16*x[1]*x[2]*x[3]*x[4]*x[5] + > x[2]^2*x[3]*x[4]*x[5] + 10*x[1]*x[3]^2*x[4]*x[5] + > 9*x[2]*x[3]^2*x[4]*x[5] + 12*x[3]^3*x[4]*x[5] + 6*x[1]*x[2]*x[4]^2*x[5] > + 6*x[2]^2*x[4]^2*x[5] + 12*x[1]*x[3]*x[4]^2*x[5] + > 16*x[2]*x[3]*x[4]^2*x[5] + 6*x[3]^2*x[4]^2*x[5] + 7*x[2]*x[4]^3*x[5] + > 15*x[3]*x[4]^3*x[5] + 7*x[1]^2*x[2]*x[5]^2 + 3*x[2]^3*x[5]^2 + > 3*x[1]^2*x[3]*x[5]^2 + 11*x[1]*x[2]*x[3]*x[5]^2 + x[2]^2*x[3]*x[5]^2 + > 12*x[1]*x[3]^2*x[5]^2 + 8*x[2]*x[3]^2*x[5]^2 + 4*x[3]^3*x[5]^2 + > x[1]*x[2]*x[4]*x[5]^2 + 14*x[2]^2*x[4]*x[5]^2 + 8*x[1]*x[3]*x[4]*x[5]^2 > + 13*x[2]*x[3]*x[4]*x[5]^2 + 4*x[3]^2*x[4]*x[5]^2 + > 15*x[2]*x[4]^2*x[5]^2 + 16*x[1]*x[2]*x[5]^3 + 10*x[2]^2*x[5]^3 + > 9*x[1]*x[3]*x[5]^3 + 14*x[2]*x[3]*x[5]^3 + 12*x[3]^2*x[5]^3 + > 10*x[2]*x[4]*x[5]^3 + x[3]*x[4]*x[5]^3 + 15*x[2]*x[5]^4 + 9*x[3]*x[5]^4, > 9*x[1]^4*x[2] + 14*x[1]^3*x[2]^2 + 5*x[1]^2*x[2]^3 + 2*x[1]*x[2]^4 + > 2*x[1]^3*x[2]*x[3] + 7*x[1]^2*x[2]^2*x[3] + 5*x[1]*x[2]^3*x[3] + > 7*x[2]^4*x[3] + 9*x[1]^2*x[2]*x[3]^2 + 12*x[1]*x[2]^2*x[3]^2 + > 2*x[2]^3*x[3]^2 + 9*x[1]*x[2]*x[3]^3 + 2*x[2]^2*x[3]^3 + x[2]*x[3]^4 + > 3*x[1]^3*x[2]*x[4] + 5*x[1]^2*x[2]^2*x[4] + 7*x[1]*x[2]^3*x[4] + > 13*x[2]^4*x[4] + 11*x[1]^2*x[2]*x[3]*x[4] + 4*x[1]*x[2]^2*x[3]*x[4] + > 11*x[2]^3*x[3]*x[4] + 14*x[1]*x[2]*x[3]^2*x[4] + 16*x[2]^2*x[3]^2*x[4] + > 15*x[1]^2*x[2]*x[4]^2 + 11*x[1]*x[2]^2*x[4]^2 + 5*x[2]^3*x[4]^2 + > 6*x[1]*x[2]*x[3]*x[4]^2 + 9*x[2]^2*x[3]*x[4]^2 + 16*x[2]*x[3]^2*x[4]^2 + > 9*x[2]^2*x[4]^3 + 15*x[2]*x[3]*x[4]^3 + 14*x[2]*x[4]^4 + 16*x[1]^4*x[5] > + 16*x[1]^3*x[2]*x[5] + 16*x[1]^2*x[2]^2*x[5] + 3*x[1]*x[2]^3*x[5] + > x[2]^4*x[5] + 9*x[1]^3*x[3]*x[5] + x[1]^2*x[2]*x[3]*x[5] + > 15*x[1]*x[2]^2*x[3]*x[5] + 10*x[2]^3*x[3]*x[5] + 7*x[1]^2*x[3]^2*x[5] + > 8*x[1]*x[2]*x[3]^2*x[5] + x[2]^2*x[3]^2*x[5] + 15*x[1]*x[3]^3*x[5] + > 5*x[2]*x[3]^3*x[5] + x[3]^4*x[5] + 11*x[1]^2*x[2]*x[4]*x[5] + > 14*x[1]*x[2]^2*x[4]*x[5] + 9*x[2]^3*x[4]*x[5] + 3*x[1]^2*x[3]*x[4]*x[5] > + 11*x[1]*x[2]*x[3]*x[4]*x[5] + 14*x[2]^2*x[3]*x[4]*x[5] + > 11*x[1]*x[3]^2*x[4]*x[5] + 13*x[2]*x[3]^2*x[4]*x[5] + 4*x[3]^3*x[4]*x[5] > + x[1]^2*x[4]^2*x[5] + 5*x[1]*x[2]*x[4]^2*x[5] + 3*x[2]^2*x[4]^2*x[5] + > 3*x[1]*x[3]*x[4]^2*x[5] + 11*x[2]*x[3]*x[4]^2*x[5] + > 12*x[3]^2*x[4]^2*x[5] + 7*x[1]*x[4]^3*x[5] + 8*x[2]*x[4]^3*x[5] + > 4*x[3]*x[4]^3*x[5] + 2*x[4]^4*x[5] + 8*x[1]^3*x[5]^2 + > 6*x[1]^2*x[2]*x[5]^2 + 11*x[1]*x[2]^2*x[5]^2 + 14*x[2]^3*x[5]^2 + > 10*x[1]^2*x[3]*x[5]^2 + 14*x[1]*x[2]*x[3]*x[5]^2 + 2*x[2]^2*x[3]*x[5]^2 > + 9*x[1]*x[3]^2*x[5]^2 + 6*x[2]*x[3]^2*x[5]^2 + 11*x[3]^3*x[5]^2 + > 4*x[1]^2*x[4]*x[5]^2 + 12*x[1]*x[2]*x[4]*x[5]^2 + 13*x[2]^2*x[4]*x[5]^2 > + 11*x[1]*x[3]*x[4]*x[5]^2 + 10*x[2]*x[3]*x[4]*x[5]^2 + > 14*x[3]^2*x[4]*x[5]^2 + 10*x[1]*x[4]^2*x[5]^2 + 4*x[2]*x[4]^2*x[5]^2 + > 5*x[3]*x[4]^2*x[5]^2 + 4*x[4]^3*x[5]^2 + 3*x[1]^2*x[5]^3 + > 13*x[1]*x[2]*x[5]^3 + 6*x[2]^2*x[5]^3 + 10*x[1]*x[3]*x[5]^3 + > 2*x[2]*x[3]*x[5]^3 + 11*x[3]^2*x[5]^3 + 3*x[1]*x[4]*x[5]^3 + > 11*x[2]*x[4]*x[5]^3 + 7*x[3]*x[4]*x[5]^3 + 3*x[4]^2*x[5]^3 + x[1]*x[5]^4 > + 9*x[2]*x[5]^4 + 5*x[3]*x[5]^4 + 6*x[4]*x[5]^4 + 16*x[5]^5, > 13*x[1]^4*x[2] + 8*x[1]^3*x[2]^2 + 14*x[1]^2*x[2]^3 + 3*x[1]*x[2]^4 + > 11*x[2]^5 + 7*x[1]^3*x[2]*x[3] + 3*x[1]^2*x[2]^2*x[3] + 12*x[2]^4*x[3] + > 3*x[1]^2*x[2]*x[3]^2 + 13*x[1]*x[2]^2*x[3]^2 + 3*x[2]^3*x[3]^2 + > 7*x[1]*x[2]*x[3]^3 + 2*x[2]^2*x[3]^3 + 7*x[2]*x[3]^4 + > 13*x[1]^3*x[2]*x[4] + 6*x[1]^2*x[2]^2*x[4] + 6*x[1]*x[2]^3*x[4] + > 6*x[2]^4*x[4] + 2*x[1]^2*x[2]*x[3]*x[4] + 15*x[1]*x[2]^2*x[3]*x[4] + > 14*x[2]^3*x[3]*x[4] + 3*x[1]*x[2]*x[3]^2*x[4] + 16*x[2]^2*x[3]^2*x[4] + > 3*x[2]*x[3]^3*x[4] + 6*x[1]^2*x[2]*x[4]^2 + 10*x[2]^3*x[4]^2 + > 7*x[2]^2*x[3]*x[4]^2 + 13*x[1]*x[2]*x[4]^3 + 5*x[2]^2*x[4]^3 + > 15*x[2]*x[3]*x[4]^3 + 13*x[2]*x[4]^4 + 16*x[1]^4*x[5] + > 5*x[1]^3*x[2]*x[5] + 11*x[1]^2*x[2]^2*x[5] + 3*x[1]*x[2]^3*x[5] + > 14*x[2]^4*x[5] + 10*x[1]^3*x[3]*x[5] + 2*x[1]^2*x[2]*x[3]*x[5] + > 14*x[1]*x[2]^2*x[3]*x[5] + 4*x[2]^3*x[3]*x[5] + 7*x[1]^2*x[3]^2*x[5] + > 10*x[1]*x[2]*x[3]^2*x[5] + 16*x[2]^2*x[3]^2*x[5] + 13*x[1]*x[3]^3*x[5] + > 2*x[2]*x[3]^3*x[5] + 3*x[3]^4*x[5] + 5*x[1]^3*x[4]*x[5] + > 7*x[1]^2*x[2]*x[4]*x[5] + 8*x[1]*x[2]^2*x[4]*x[5] + 2*x[2]^3*x[4]*x[5] + > 16*x[1]^2*x[3]*x[4]*x[5] + 9*x[1]*x[2]*x[3]*x[4]*x[5] + > 15*x[1]*x[3]^2*x[4]*x[5] + 3*x[2]*x[3]^2*x[4]*x[5] + 5*x[3]^3*x[4]*x[5] > + 10*x[1]^2*x[4]^2*x[5] + 10*x[2]^2*x[4]^2*x[5] + x[1]*x[3]*x[4]^2*x[5] > + x[2]*x[3]*x[4]^2*x[5] + 15*x[3]^2*x[4]^2*x[5] + 15*x[1]*x[4]^3*x[5] + > 14*x[2]*x[4]^3*x[5] + 3*x[3]*x[4]^3*x[5] + 13*x[4]^4*x[5] + > 4*x[1]^3*x[5]^2 + 13*x[1]^2*x[2]*x[5]^2 + 16*x[1]*x[2]^2*x[5]^2 + > 14*x[2]^3*x[5]^2 + 3*x[1]^2*x[3]*x[5]^2 + 16*x[1]*x[2]*x[3]*x[5]^2 + > 11*x[2]^2*x[3]*x[5]^2 + 8*x[1]*x[3]^2*x[5]^2 + 10*x[2]*x[3]^2*x[5]^2 + > x[3]^3*x[5]^2 + 5*x[1]^2*x[4]*x[5]^2 + 15*x[1]*x[2]*x[4]*x[5]^2 + > 9*x[2]^2*x[4]*x[5]^2 + 10*x[1]*x[3]*x[4]*x[5]^2 + > 9*x[2]*x[3]*x[4]*x[5]^2 + 12*x[3]^2*x[4]*x[5]^2 + 12*x[1]*x[4]^2*x[5]^2 > + 3*x[2]*x[4]^2*x[5]^2 + 6*x[3]*x[4]^2*x[5]^2 + 15*x[4]^3*x[5]^2 + > 3*x[1]^2*x[5]^3 + 10*x[1]*x[2]*x[5]^3 + 14*x[2]^2*x[5]^3 + > 12*x[1]*x[3]*x[5]^3 + 6*x[2]*x[3]*x[5]^3 + 4*x[3]^2*x[5]^3 + > 8*x[1]*x[4]*x[5]^3 + 4*x[3]*x[4]*x[5]^3 + 9*x[1]*x[5]^4 + 14*x[2]*x[5]^4 > + 12*x[3]*x[5]^4 + x[4]*x[5]^4 + 9*x[5]^5]);
We check a few of the invariants of \(X\).
> Dimension(X); 2 > IsNonsingular(X); true > ArithmeticGenus(X); 0 > // Get the sectional genus of X -- ie the genus of a hyperplane section. > ArithmeticGenus(X meet Scheme(P,P.1)); 9
Now we construct the canonical sheaf and hyperplane sheaf and check intersection numbers.
> KX := CanonicalSheaf(X); > HX := StructureSheaf(X,1); // hyperplane sheaf > IntersectionPairing(HX,HX); // should be 10 = Degree(X) 10 > Degree(X); 10 > IntersectionPairing(KX,HX); // should be 6 6 > IntersectionPairing(KX,KX); // should be -9 : lots of exceptional curves! -9
We now get the adjunction map as a divisor map, compute its image \(X_1\) and check some of the invariants of \(X_1\) as well as its corresponding intersection numbers.
> mp1,X1 := DivisorMap(Twist(KX,1)); > Dimension(Ambient(X1)); Dimension(X1); 8 2 > KX1 := CanonicalSheaf(X1); > HX1 := StructureSheaf(X1,1); // hyperplane sheaf of X1 > IntersectionPairing(HX1,HX1); // should be 13 = degree X1 13 > IntersectionPairing(KX1,HX1); // should be -3 -3 > IntersectionPairing(KX1,KX1); // should be -2 : fewer exceptional curves! -2
We construct a second adjunction map to get \(X_2\) and check it as above.
> mp2,X2 := DivisorMap(Twist(KX1,1)); > Dimension(Ambient(X2)); Dimension(X2); 5 2 > KX2 := CanonicalSheaf(X2); > HX2 := StructureSheaf(X2,1); // hyperplane sheaf X2 > IntersectionPairing(HX2,HX2); // = degree X2 = 5 5 > IntersectionPairing(KX2,HX2); // should be -5 -5 > IntersectionPairing(KX2,KX2); // should be 5 5
Now \(X_2\) should be a degree five Del Pezzo surface with \({\cal K}_X \simeq {{\cal O}_X}(-1)\). This last isomorphism can be verified by checking that there is a degree \(-2\) isomorphism from \({\cal K}_X\) to \({{\cal O}_X}(1)\)! The scheme \(X_2\) is much simpler than \(X\): it is defined by five degree 2 polynomials.
> boo,d := IsIsomorphicWithTwist(KX2,HX2); > boo; d; true -2 > MinimalBasis(Ideal(X2)); Scheme over GF(17) defined by y[1]^2 + y[3]^2 + y[1]*y[4] + 15*y[2]*y[4] + 8*y[3]*y[4] + 6*y[4]^2 + 2*y[1]*y[5] + 12*y[2]*y[5] + y[3]*y[5] + 4*y[4]*y[5] + 4*y[5]^2 + 6*y[1]*y[6] + 10*y[2]*y[6] + 7*y[3]*y[6] + 7*y[5]*y[6] + 16*y[6]^2, y[1]*y[2] + 13*y[3]^2 + 3*y[1]*y[4] + 14*y[2]*y[4] + 13*y[3]*y[4] + 5*y[4]^2 + 14*y[1]*y[5] + 10*y[2]*y[5] + 2*y[3]*y[5] + 9*y[4]*y[5] + 6*y[5]^2 + 4*y[1]*y[6] + 13*y[2]*y[6] + 10*y[3]*y[6] + 3*y[4]*y[6] + y[5]*y[6] + 12*y[6]^2, y[2]^2 + 16*y[3]^2 + 15*y[1]*y[4] + 3*y[3]*y[4] + y[4]^2 + 10*y[1]*y[5] + 12*y[2]*y[5] + 10*y[3]*y[5] + 11*y[4]*y[5] + 9*y[5]^2 + 5*y[1]*y[6] + 3*y[2]*y[6] + 2*y[3]*y[6] + 15*y[4]*y[6] + 12*y[5]*y[6] + 5*y[6]^2, y[1]*y[3] + 13*y[3]^2 + y[1]*y[4] + 11*y[3]*y[4] + y[4]^2 + 16*y[1]*y[5] + y[2]*y[5] + 15*y[3]*y[5] + 3*y[4]*y[5] + 7*y[1]*y[6] + 3*y[2]*y[6] + 9*y[3]*y[6] + 10*y[4]*y[6] + 8*y[5]*y[6] + 6*y[6]^2, y[2]*y[3] + 16*y[3]^2 + 14*y[1]*y[4] + 3*y[2]*y[4] + y[3]*y[4] + y[4]^2 + 12*y[1]*y[5] + 9*y[3]*y[5] + 6*y[4]*y[5] + 2*y[5]^2 + 13*y[3]*y[6] + 9*y[4]*y[6] + 13*y[5]*y[6] + 12*y[6]^2
Finally we get the composed map from \(X\) to \(X_2\) and check that it is (birationally) invertible.
> mp1r := Restriction(mp1,X,X1); > mp2r := Restriction(mp2,X1,X2); > mpc := Expand(mp1r*mp2r); > boo := IsInvertible(mpc); > boo; true
- Example: Ell Crv Norm (ex-362695)#
In this example, we show how the sheaf machinery can be effectively used as an alternative method to normalise the projective coordinate ring of a normal, but not projectively normal, projective variety. Here the coordinate ring is locally normal at all primes except at the maximal homogeneous ideal.
Our chosen variety is \(C\), an elliptic curve that has been embedded as a degree 8 subvariety of \({\mathbb{P}}^3\) over \({\bf Q}\). The curve \(C\) can be thought of as having been embedded in \({\mathbb{P}}^7\) by a complete linear system of degree 8 and then (isomorphically) projected down into \({\mathbb{P}}^3\). Such genus one curves embedded as degree 8 curves in \({\mathbb{P}}^3\) actually arise fairly naturally as models of homogeneous spaces arising in eight-descents.
We wish to recover the full embedding as a projective normal curve in \({\mathbb{P}}^7\). The coordinate ring of this is isomorphic to the normalisation of the coordinate ring of \(C\) in \({\mathbb{P}}^3\). From a sheaf-theoretic point of view, this is straightforward. The full embedding is the image of the divisor map corresponding to a hyperplane section of \(C\) or, equivalently, to the Serre twisting sheaf \({{\cal O}_X}(1)\). The maximal module of \({{\cal O}_X}(1)\) is isomorphic to the normalisation as an \(R\)-module, where \(R\) is the coordinate ring of \(C\) in \({\mathbb{P}}^3\), and it can be recovered as an algebra by taking the image of its associated divisor map. The global sections of \({{\cal O}_X}(1)\) correspond to the full Riemann-Roch space of the divisor on the abstract curve given by a certain hyperplane divisor on \(C\).
This example also illustrates another interesting point. In situations similar to these, the dimension of the full space of global sections of the Serre twisting sheaf can be computed from cohomology of the coordinate ring \(R\). However, it is faster in this case to explicitly compute the full maximal module of \({{\cal O}_X}(1)\), the zero-th graded part of this corresponding to the space of global sections and having the dimension of the zeroth cohomology group. In fact, though we only need to compute the dimension of this part, it is actually much quicker to compute the maximal module and compute its cohomology than to compute the cohomology of the original defining module, which is \(R\) twisted once. This probably reflects to some extent the fact that polynomial ring Groebner basis computations are much more highly tuned currently in Magma than the alternating algebra ones used in the cohomology computations. But the maximal module of a sheaf is generally a nicer object than a submodule with bits missing in the lower-graded pieces and has a smaller Castelnuevo-Mumford regularity etc. So, as we see in this example, it is often worth making sure that the maximal module of a sheaf is available before making cohomology calls.
> P<x,y,z,t> := ProjectiveSpace(Rationals(),3); > C := Curve(P,[ x^2*y^2 - 23/59*x*y^3 + 9/59*y^4 + 27/59*x^3*z - 23/59*x^2*y* > z - 6/59*x*y^2*z + 6/59*y^3*z - 10/59*x^2*z^2 + 5/59*x*y*z^2 - 3/59*y^2*z^2 + > 1/59*x*z^3 - 74/59*x^3*t + 115/59*x^2*y*t - 83/59*x*y^2*t + 3/59*y^3*t - > 105/59*x^2*z*t + 1/59*x*y*z*t - 2/59*y^2*z*t + 36/59*x*z^2*t + 4/59*y*z^2*t - > 3/59*z^3*t + 297/59*x^2*t^2 - 135/59*x*y*t^2 + 52/59*y^2*t^2 + 68/59*x*z*t^2 - > 11/59*y*z*t^2 - 18/59*z^2*t^2 - 315/59*x*t^3 + 42/59*y*t^3 + 96/59*t^4, > x^3*y - 833/354*x*y^3 - 11/236*y^4 - 1633/708*x^3*z - 4675/708*x^2*y*z - > 2633/708*x*y^2*z - 27/236*y^3*z + 805/354*x^2*z^2 + 223/59*x*y*z^2 - > 4/59*y^2*z^2 - 38/59*x*z^3 + 3359/708*x^3*t + 3811/354*x^2*y*t + > 1445/708*x*y^2*t + 303/118*y^3*t - 715/177*x^2*z*t - 527/177*x*y*z*t + > 211/118*y^2*z*t + 347/354*x*z^2*t - 195/236*y*z^2*t - 4/59*z^3*t - > 127/236*x^2*t^2 - 8237/708*x*y*t^2 + 65/708*y^2*t^2 + 1973/708*x*z*t^2 + > 123/59*y*z*t^2 - 24/59*z^2*t^2 - 1753/354*x*t^3 + 873/236*y*t^3 + 128/59*t^4, > x^4 + 269/354*x*y^3 + 35/236*y^4 + 1849/708*x^3*z + 4255/708*x^2*y*z - > 247/708*x*y^2*z + 43/236*y^3*z - 727/354*x^2*z^2 - 82/59*x*y*z^2 + > 2/59*y^2*z^2 + 19/59*x*z^3 - 5603/708*x^3*t - 3469/354*x^2*y*t - > 1637/708*x*y^2*t - 63/118*y^3*t + 328/177*x^2*z*t - 769/177*x*y*z*t - > 17/118*y^2*z*t + 151/354*x*z^2*t + 127/236*y*z^2*t + 2/59*z^3*t + > 1391/236*x^2*t^2 + 7865/708*x*y*t^2 + 823/708*y^2*t^2 - 1901/708*x*z*t^2 + > 86/59*y*z*t^2 + 12/59*z^2*t^2 + 493/354*x*t^3 - 761/236*y*t^3 - 64/59*t^4]);
Next the hyperplane sheaf of \(C\) is constructed and the dimension of the space of global sections is confirmed to be 8 using
DimensionOfGlobalSections(which also saturates the sheaf).> OC1 := StructureSheaf(C,1); > DimensionOfGlobalSections(OC1); 8
Finally, the projective normal embedding into \(P^7\) is created and we check that the image \(X\) is defined by 20 quadrics.
> norm_mp, X := DivisorMap(OC1); > ArithmeticGenus(X); 1 > B := MinimalBasis(Ideal(X)); > #B; 20 > [TotalDegree(f) : f in B]; [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]