Vector spaces of singular planar cubics - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T09:27:14Z http://mathoverflow.net/feeds/question/3300 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/3300/vector-spaces-of-singular-planar-cubics Vector spaces of singular planar cubics Daniel Erman 2009-10-29T17:47:48Z 2009-10-30T02:34:40Z <p>What is the largest dimensional linear space of singular planar cubics? Is this known?</p> <p>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.</p> http://mathoverflow.net/questions/3300/vector-spaces-of-singular-planar-cubics/3314#3314 Answer by jvp for Vector spaces of singular planar cubics jvp 2009-10-29T18:55:16Z 2009-10-30T02:34:40Z <p>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.</p> <p><strong>Edit</strong> As David pointed out in another answer, this argument uses char 0.</p> http://mathoverflow.net/questions/3300/vector-spaces-of-singular-planar-cubics/3316#3316 Answer by David Speyer for Vector spaces of singular planar cubics David Speyer 2009-10-29T19:08:01Z 2009-10-29T19:08:01Z <p>The answer of jvp (above) is completely right in characteristic zero. </p> <p>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. </p> <p>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. </p>