#8.5
Let C be a nonsingular, irreducible cubic. Prove that any flex of C lies on the tangents of either one or three other points of C. Moreover, if a flex lies on the tangents of three other points of C, prove that the three points are collinear. (the book then hints to see Thms 8.1-8.3)
Thm. 8.1
(i) A cubic is tangent to the line at infinity at the point at infinity on vertical lines, has a flex there, and is irreducible iff it has equation y2+cxy+dy=ex3+fx2+gx+h for real numbers e-h with e not equal to zero.
(ii) A cubic C is irreducible and has a flex at a point P iff there is a transformation that takes C to y2=x3+fx2+gx+h for real numbers f, g, h and takes P to the point at infinity on vertical lines.
Thm. 8.2
Let C be the cubic y2-q(x) for q(x)=x3+fx2+gx+h
(i) Then C is nonsingular at all of its points in the Euclidean plane that do not lie on the x-axis, and the tangents at these points are not vertical.
(ii) A point (r,0) on the x-axis in the Euclidean plane lies on C iff x-r is a factor of q(x). If x-r is not a repeated factor of q(x), then C is nonsingular at (r,0) and has a vertical tangent there. If x-r is a repeated factor of q(x), then C is singular at (r,0)
(iii) The one point of C at infinity is the point at infinity on vertical lines, and C is nonsingular there and tangent to the line at infinity.
Thm. 8.3
A cubic is nonsingular and irreducible and has a flex iff it can be transformed into y2=x(x-1)(x-w) or y2=x(x2+kx+1) for w>1 and -2<k<2.
(from Conics and Cubics: A Concrete Introduction to Algebraic Curves by Robert Bix)
I am guessing that I want to use a tranformation that takes C to the one in Thm. 8.1(ii) and perhaps one of those in Thm. 8.3, but I'm unsure of what direction to go from there. Proofs are not a strong suit of mine at this point. The x-intercepts are where the transformed curve would then have tangent lines that are vertical by Thm. 8.2, and certainly points that lie on the x-axis are collinear (and transformations preserve collinearity). Also, I know the transformed cubic intersect the x-axis at least once. Anyway, any further clarification would be appreciated.
Let C be a nonsingular, irreducible cubic. Prove that any flex of C lies on the tangents of either one or three other points of C. Moreover, if a flex lies on the tangents of three other points of C, prove that the three points are collinear. (the book then hints to see Thms 8.1-8.3)
Thm. 8.1
(i) A cubic is tangent to the line at infinity at the point at infinity on vertical lines, has a flex there, and is irreducible iff it has equation y2+cxy+dy=ex3+fx2+gx+h for real numbers e-h with e not equal to zero.
(ii) A cubic C is irreducible and has a flex at a point P iff there is a transformation that takes C to y2=x3+fx2+gx+h for real numbers f, g, h and takes P to the point at infinity on vertical lines.
Thm. 8.2
Let C be the cubic y2-q(x) for q(x)=x3+fx2+gx+h
(i) Then C is nonsingular at all of its points in the Euclidean plane that do not lie on the x-axis, and the tangents at these points are not vertical.
(ii) A point (r,0) on the x-axis in the Euclidean plane lies on C iff x-r is a factor of q(x). If x-r is not a repeated factor of q(x), then C is nonsingular at (r,0) and has a vertical tangent there. If x-r is a repeated factor of q(x), then C is singular at (r,0)
(iii) The one point of C at infinity is the point at infinity on vertical lines, and C is nonsingular there and tangent to the line at infinity.
Thm. 8.3
A cubic is nonsingular and irreducible and has a flex iff it can be transformed into y2=x(x-1)(x-w) or y2=x(x2+kx+1) for w>1 and -2<k<2.
(from Conics and Cubics: A Concrete Introduction to Algebraic Curves by Robert Bix)
I am guessing that I want to use a tranformation that takes C to the one in Thm. 8.1(ii) and perhaps one of those in Thm. 8.3, but I'm unsure of what direction to go from there. Proofs are not a strong suit of mine at this point. The x-intercepts are where the transformed curve would then have tangent lines that are vertical by Thm. 8.2, and certainly points that lie on the x-axis are collinear (and transformations preserve collinearity). Also, I know the transformed cubic intersect the x-axis at least once. Anyway, any further clarification would be appreciated.
