I understand that a cubic curve in $\mathbb P^2$ is given by a homogeneous degree $3$ polynomial, so that the space of cubic curves can be identified with $\mathbb P^9$.
However, there should be nine types of cubic curves:
- irreducible (elliptic) curves which may be either smooth, with a nodal singularity or a cusp singularity (3 types)
- the union of a conic and a line, which may either be tangent to the conic or meet the conic in two distinct points (2 types)
- the union of three distinct lines, which may either meet in one point or in three points (2 types)
- the union of a double line with a line (1 type)
- a triple line (1 type)
I guess that $\mathbb P^9$ should admit a stratification by "type", but what are the dimensions of the components. Are there $0$-dimensional components?
What is a good reference for this stratification?
Best Answer
We'll make the standard assumptions about working over an algebraically closed field of characteristic not $2$ or $3$.
First, the locus of irreducible cubics is open with complement of codimension $2$: those curves which can be represented as the product of a linear term and a quadratic term lie in the image of an embedded $\Bbb P^2\times \Bbb P^5$ inside $\Bbb P^9$. We can then put the irreducible cubics in Weierstrass form, which after dehomogenizing with respect to $z$ looks like $y^2=g(x)$ for $g$ a degree three polynomial. The curve will be smooth if all the roots of $g$ are distinct, have a node if there is a single root and a double root, and have a cusp if there is a triple root. The first condition is open, as it's represented by the nonvanishing of $Res_x(g,g')$, while the second is codimension-one, as it's given by the vanishing of $Res_x(g,g')$ and the nonvanishing of $Res_x(g',g'')$, while the last is codimension-two, given by the vanishing of $Res_x(g,g')$ and $Res_x(g',g'')$.
Now for those curves who's equation decomposes as a product of a linear factor and a quadratic factor. If the quadratic factor is reducible to a product of linear factors, then this curve lies in the image of $\Bbb P^2\times \Bbb P^2\times\Bbb P^2$ inside $\Bbb P^2\times \Bbb P^5$, so the locus of curve which corresponds to a line and a nondegenerate conic is of dimension $7$. Here, the condition that the intersection points of the line and conic should be distinct is an open condition, while the case of a conic and it's tangent line cuts down the space of possibilities by 1 via requiring the two intersection points of the line and conic be equal. So the first is seven-dimensional and the latter is six-dimensional.
For those curves who's equation factors as a product of three linear terms, we see that they are in the image of a $(\Bbb P^2)^3$. The condition that all three lines are distinct and do not share a common intersection point is an open condition and thus these form a 6-dimensional set. The locus of three distinct lines meeting in a point is cut out by the one condition that the intersection points $\ell_1\cap \ell_2$ and $\ell_1\cap \ell_3$ coincide, so this is a 5-dimensional set.
If we have two lines, one of which is fat, then this corresponds to the complement of the diagonal of $\Bbb P^2\times \Bbb P^2$, embedded in $\Bbb P^9$ and is thus 4-dimensional. For a triple line, this is exactly specified by the equation of the line, and is thus a $\Bbb P^2$ worth of options, so it's two-dimensional.
To summarize, the locii are of the following form: