I have a very concrete question about degree $d$ curves in $\mathbb{P}^2$. Let $$\mathcal{D} \approx \mathbb{P}^{\delta_d}$$ be the space of homogeneous degree $d$ polynomials in three variables upto scaling, where $\delta_d = \frac{d(d+3)}{2}$. Furthermore, let $$ \mathcal{D}(r) \approx \mathbb{P}^r \subset \mathcal{D}$$ be the space of degree $d$ polynomials passing through $\delta_d-r$ points in $\textit{general}$ position. Note that the dimension of this space is $r$. The question I have is as follows......let $$f\in \mathcal{D}(7),$$ i.e. let $f$ be a degree $d$ polynomial passing through $\delta_d -7$ points in general position. Is it correct that the only possible singularities that $f$ can have are $A_k$ for $k= 1$ to $7$, $D_k$ for $k=4$ to $7$ and $E_6$ and $E_7$. The reason I have in mind is that upto codimension $7$
the only possible singularities are the ADE singularities. Any other singularity I want to rule out by saying that the points are in ``general position''. Is this argument correct?

  • $\begingroup$ The idea looks right, but it is not a proof so far. The codimension of singularity types you refer to does not interact so nicely with assumptions on degrees. (For instance, a non-isolated singularity of type "$A_\infty$" has infinite codimension in your sense, but it has codimension 3 for $d=2$). It is possible that your statement (modified as "the only possible isolated singularities are the ADE singularities") is right, but a priori it is conceivable that it is true only for degrees bounded from below, i.e., that there are a finite number of exceptions. Do you have any application in mind? $\endgroup$ – quim Oct 20 '11 at 20:31
  • $\begingroup$ I am assuming that d is ``sufficiently'' large. I forgot to say that. $\endgroup$ – Ritwik Oct 21 '11 at 4:54

Nonreduced curves in $|\mathcal{O}(d)|$ occur in codimension $2d-1$, so you will want $d > 4$ to get rid of them.

Fix a topological singularity type, and let $s$ be the codimension of its equisingular family inside its versal deformation. Then in $\mathcal{O}(d)$, singularities of this kind will occur in codimension at least $\mathrm{min}(d+1, s)$ because any length $d+1$ subscheme imposes indepedent conditions on $\mathcal{O}(d)$, and so the former can be shown to be transverse to a general $d+1$ codimensional subspace of the versal deformation of a singularity. Thus once $d \ge 7$, any singularity with $s > 8$ will not appear. This certainly limits the ADE singularities which may appear to the ones you listed. And if you have a handy table saying the only singularities with $s \le 7$ are ADE, then you could conclude.

Because I don't have such a table (and because I'm trying to stay awake long enough to get over my jetlag), let's show it. Certainly $s \ge \delta$, and equality holds only for a node. So you only have to worry about singularities which have $\delta < 7$. In fact there's a better estimate coming from considering the 'equiclassical locus': $s \ge \kappa = \delta + \sum(m_i -1)$ where $m_i$ are the multiplicities of the components.

Recall that $\delta$ is the sum over all infinitely near points of $m(m-1)/2$, where $m$ is the multiplicity of the curve at that point. In particular, if $\delta \le 6$, then certainly $m \le 4$, and moreover if the multiplicity is $4$ then all branches must be smoothed and made disjoint after one blowup. Now I list topological types of singularities with $\delta < 7$, by number of branches:

  1. $y^2 = x^{2n+1}$ for $n \le 6$, or $y^3 = x^4, x^5, x^7$, or $x^4 = y^5$.
  2. $y^2 = x^{2n}$ for $n \le 6$, or a union of $y^2 = x^{2n+1}$ with a transverse line for $n \le 4$, the union of $y^2 = x^3$ with its tangent cone, or two ordinary cusps $(y^2 = x^3)$ with different tangent directions.
  3. If any branches are singular, it must be the union of an ordinary cusp with two transverse lines. One can have $x(y^2 - x^{2n})$ for $n = 1, 2, 3$. Also there is the possibility $y(y+ax^2)(y+bx^2)$.
  4. Four mutually transverse lines.

How did I make this list? Unibranch singularities are classified by sub semigroups of $\mathbb{N}$, so I listed the ones corresponding to semigroups with at most 6 gaps; then I tried gluing them together. Most of the above list were ADE singularities. The ones that weren't:

(a) $x^3 = y^7$ (b) $x^4 = y^5$, (c) two ordinary cusps with different tangents, (d) an ordinary cusp with two transverse lines, (e) $y(y+ax^2)(y+bx^2)$, and (f) four mutually transverse lines.

The items (a), (b), (c), are ruled out by the constraint $\kappa < 8$. In the other cases, there is a obvious $\delta$-constant deformation (move a smooth branch) which on the one hand changes the topological type of the singularity, and on the other does not reduce it to all nodes, so the equisingular locus must be of codimension at least 2 in the $\delta$-constant locus.

One reference for this kind of stuff is Diaz and Harris's ``Ideals associated to deformations of singular plane curves''.

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