// Nonsingularity assertions of Theorem 1.1. // (0) The 9-fold V has singular locus // AA^4 union the 3 planes AA^2 RR := PolynomialRing(Rationals(),[2,2,2,3,1,1,1,2,2,2,3,3,3]); L := [ S*x0*x2 + r0*x2*y0 + r2*x0*y2 - y1*z1, -r1*r2*x0^2 - S*x0*y0 - r0*y0^2 + z1*z2, S*x0*x1 + r0*x1*y0 + r1*x0*y1 - y2*z2, -r1*x0*x2 + y0*y2 - x1*z1, r2*x0*x1 - y0*y1 + x2*z2, -(-r0*x1*x2 + y1*y2 - x0*z0), -(r0*r2*x1^2 + S*x1*y1 + r1*y1^2 - z0*z2), -(S*x1*x2 + r1*x2*y1 + r2*x1*y2 - y0*z0), -(r0*r1*x2^2 + S*x2*y2 + r2*y2^2 - z0*z1) ]; J := JacobianMatrix(L); RRbar := PolynomialRing(Rationals(),[2,2,2,3,1,1,1]); fie := hom RRbar | r0,r1,r2,S,x0,x1,x2,0,0,0,0,0,0>; Jbar := Matrix(13,[ fie(J[i,j]) : j in [1..13], i in [1..9] ]); V := ReducedSubscheme(Scheme(Spec(RRbar), fie(L) cat Minors(Jbar,4))); Dimension(V); X := IrreducibleComponents(V); X[1]; [Dimension(x) : x in X]; X[2]; // (A) The S3 symmetric surface Y univeral cover of the // ZZ/3 Godeaux is nonsingular, and ZZ/3 action is free RR := PolynomialRing(Rationals(),[1,1,2,2,2,3,3]); x0 := -x1 - x2; z0 := -z1 - z2; r0 := y0 + x0^2 + 7*x1*x2; r1 := y1 + x1^2 + 7*x0*x2; r2 := y2 + x2^2 + 7*x0*x1; s := x0^3 + x1^3 + x2^3; L := [ s*x0*x2 + r0*x2*y0 + r2*x0*y2 - y1*z1, -r1*r2*x0^2 - s*x0*y0 - r0*y0^2 + z1*z2, s*x0*x1 + r0*x1*y0 + r1*x0*y1 - y2*z2, -r1*x0*x2 + y0*y2 - x1*z1, r2*x0*x1 - y0*y1 + x2*z2, -(-r0*x1*x2 + y1*y2 - x0*z0), -(r0*r2*x1^2 + s*x1*y1 + r1*y1^2 - z0*z2), -(s*x1*x2 + r1*x2*y1 + r2*x1*y2 - y0*z0), -(r0*r1*x2^2 + s*x2*y2 + r2*y2^2 - z0*z1) ]; J := JacobianMatrix(L); JJ := Minors(J,4); time z0^4 in Ideal(JJ); // true. Time = 1.3 secs time y0^5 in Ideal(JJ cat [z0,z1,z2]); // true. Time = 0.8 secs time x0^13 in Ideal(JJ cat [z0,z1,z2,y0,y1,y2]); // true 0.8 secs // The ZZ/3 action on Y is fixed point free Fix := L cat [x1^3-x0^3, x2^3-x1^3, y1^3-y0^3, y2^3-y1^3, z1-z0, z2-z1]; Dimension(Scheme(Proj(RR), Fix)); // gives -1, the empty set time y0^4 in Ideal(Fix); // true 0 secs time x0^8 in Ideal(Fix); // true 0 secs // (B) The S3 symmetric 4-fold F of index 3 is quasismooth RR := PolynomialRing(Rationals(),[1,1,1,2,2,2,3,3,3]); r0 := y0 + x0^2 + 7*x1*x2; r1 := y1 + x1^2 + 7*x0*x2; r2 := y2 + x2^2 + 7*x0*x1; s := x0^3 + x1^3 + x2^3; L := [ s*x0*x2 + r0*x2*y0 + r2*x0*y2 - y1*z1, -r1*r2*x0^2 - s*x0*y0 - r0*y0^2 + z1*z2, s*x0*x1 + r0*x1*y0 + r1*x0*y1 - y2*z2, -r1*x0*x2 + y0*y2 - x1*z1, r2*x0*x1 - y0*y1 + x2*z2, -(-r0*x1*x2 + y1*y2 - x0*z0), -(r0*r2*x1^2 + s*x1*y1 + r1*y1^2 - z0*z2), -(s*x1*x2 + r1*x2*y1 + r2*x1*y2 - y0*z0), -(r0*r1*x2^2 + s*x2*y2 + r2*y2^2 - z0*z1) ]; J := JacobianMatrix(L); JJ := Minors(J,4); time z0^4 in Ideal(JJ); // true. Time = 11 secs time y0^5 in Ideal(JJ cat [z0,z1,z2]); // true. Time = 11 secs time x0^13 in Ideal(JJ cat [z0,z1,z2,y0,y1,y2]); // true 10 secs // (C) The S3 symmetric Calabi-Yau is nonsingular RR := PolynomialRing(Rationals(),[1,1,1,2,2,2,3,3]); z0 := -z1 - z2; r0 := y0 + x0^2 + 7*x1*x2; r1 := y1 + x1^2 + 7*x0*x2; r2 := y2 + x2^2 + 7*x0*x1; s := x0^3 + x1^3 + x2^3; L := [ s*x0*x2 + r0*x2*y0 + r2*x0*y2 - y1*z1, -r1*r2*x0^2 - s*x0*y0 - r0*y0^2 + z1*z2, s*x0*x1 + r0*x1*y0 + r1*x0*y1 - y2*z2, -r1*x0*x2 + y0*y2 - x1*z1, r2*x0*x1 - y0*y1 + x2*z2, -(-r0*x1*x2 + y1*y2 - x0*z0), -(r0*r2*x1^2 + s*x1*y1 + r1*y1^2 - z0*z2), -(s*x1*x2 + r1*x2*y1 + r2*x1*y2 - y0*z0), -(r0*r1*x2^2 + s*x2*y2 + r2*y2^2 - z0*z1) ]; J := JacobianMatrix(L); JJ := Minors(J,4); time z0^4 in Ideal(JJ); // true. Time = 6 secs time y0^5 in Ideal(JJ cat [z0,z1,z2]); // true. Time = 4 secs time x0^13 in Ideal(JJ cat [z0,z1,z2,y0,y1,y2]); // true 3 secs // (D) The S3 symmetric Fano of index 2 is quasismooth RR := PolynomialRing(Rationals(),[1,1,1,2,2,2,3,3]); x0 := -x1 - x2; r0 := y0 + x0^2 + 7*x1*x2; r1 := y1 + x1^2 + 7*x0*x2; r2 := y2 + x2^2 + 7*x0*x1; s := x0^3 + x1^3 + x2^3; L := [ s*x0*x2 + r0*x2*y0 + r2*x0*y2 - y1*z1, -r1*r2*x0^2 - s*x0*y0 - r0*y0^2 + z1*z2, s*x0*x1 + r0*x1*y0 + r1*x0*y1 - y2*z2, -r1*x0*x2 + y0*y2 - x1*z1, r2*x0*x1 - y0*y1 + x2*z2, -(-r0*x1*x2 + y1*y2 - x0*z0), -(r0*r2*x1^2 + s*x1*y1 + r1*y1^2 - z0*z2), -(s*x1*x2 + r1*x2*y1 + r2*x1*y2 - y0*z0), -(r0*r1*x2^2 + s*x2*y2 + r2*y2^2 - z0*z1) ]; J := JacobianMatrix(L); JJ := Minors(J,4); time z0^4 in Ideal(JJ); // true. Time = 2.4 secs time y0^5 in Ideal(JJ cat [z0,z1,z2]); // true. Time = 2.2 secs time x0^13 in Ideal(JJ cat [z0,z1,z2,y0,y1,y2]); // true 2.2 secs