According to Bertini's Theorem a linear system is smooth away from its base points. Thus there is a point in PP^2 contained in the singular set of every cubic in your linear system. You need three conditions to impose a singularity a point p (f(p) = f_x(p) = f_y(p)=0). Thus maximal projective spaces inside the discriminant have dimension 6.

Edit As David pointed out in another answer, this argument uses char 0.

According to Bertini's Theorem a linear system is smooth away from its base points. Thus there is a point in PP^2 contained in the singular set of every cubic in your linear system. You need three conditions to impose a singularity at a point p (f(p) = f_x(p) = f_y(p)=0). Thus maximal projective spaces inside the discriminant have dimension 6.

Edit As David pointed out in another answer, this argument uses char 0.

According to Bertini's Theorem a linear system is smooth away from its base points. Thus there is a point in PP^2 contained in the singular set of every cubic in your linear system. You need three conditions to impose a singularity a point p (f(p) = f_x(p) = f_y(p)=0). Thus maximal projective spaces inside the discriminant have dimension 6.

According to Bertini's Theorem a linear system is smooth away from its base points. Thus there is a point in PP^2 contained in the singular set of every cubic in your linear system. You need three conditions to impose a singularity a point p (f(p) = f_x(p) = f_y(p)=0). Thus maximal projective spaces inside the discriminant have dimension 6.

Edit As David pointed out in another answer, this argument uses char 0.