What is the largest dimensional linear space of singular planar cubics? Is this known?
Think of the space of planar cubics as a PP^9 (parametrized by the coefficients). The discriminant \Delta is then a degree 12 polynomial on PP^9 whose vanishing parametrizes singular cubic curves. What is the dimension of the largest linear space inside the vanishing of \Delta? I attempted doing this on a computer, but the computations were too large to terminate.
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.
The answer of jvp (above) is completely right in characteristic zero.
I'll be annoying and point out the following example, due to Serre: let K be an algebraically closed field of characteristic 2 and let F be the field with two elements inside K. Let L be the vector space of cubics in K[x,y,z] that vanish at the 7 points of P^2(F). Every cubic in this family is singular, yet there is no point at which they are all singular.
Now, L has dimension 3 (and its projectivization has dimension 2), so this is still smaller than the 7 (projectively 6) dimensional spaces of cubics which are singular at a given point. But it is a warning about the kind of issues that can come up when you ask this question in finite characteristic.
