TITLE: Cubic plane curves
QUESTION [6 upvotes]: 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?

REPLY [5 votes]: 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:

Smooth cubic: open $9$-dimensional
Nodal cubic: open subset of $8$-dimensional
Cuspidal cubic: open subset of $7$-dimensional
Nondegenerate conic and secant: open subset of $7$-dimensional
Nondegenerate conic and tangent: open subset of $6$-dimensional
Three generic lines ("triangle"): open subset of $6$-dimensional
Three lines meeting at a point ("asterisk"): open subset of $5$-dimensional
Two lines, one fat: open subset of $4$-dimensional
Triple line: closed $2$-dimensional