Curves with prescribed singularities

It is a classical and interesting problem, which is still in the centre of theoretical research, to study the variety $V = V_d(S_1, \dots, S_r)$ of (irreducible) curves $C \subset \P ^2_\mathbf{C}$ of degree $d$ having exactly $r$ singularities of prescribed (topological or analytical) types $S_1, \dots, S_r$. Among the most important questions are:

• Is $V \not= \emptyset$ (existence problem)?
• Is $V$ irreducible (irreducibility problem)?
• Is $V$ smooth of expected dimension ($T$-smoothness problem)?

A complete answer is only known in the special case of nodal curves, that is, for $V_d(r) = V_d(S_1, \dots, S_r)$ with $S_i$ ordinary nodes ($A_1$-singularities): $V_d(r) \not= \emptyset$ and $T$-smooth $\Longleftrightarrow r \le \frac{(d-1)(d-2)}{2}$ (Severi, 1921), $V_d(r)$ is irreducible (if $\not= \emptyset$) (Harris, 1985). Even for cuspidal curves there is no sufficient and necessary answer to any of the above questions and one can hardly expect such an answer.

Clearly, one can easily give an upper bound for the number of singular points that may occur on a plane irreducible curve $C$ of degree $d$: by the genus formula $C$ can have, at most, $(d\:-\:1)(d\:-\:2)/2$ singularities. Another upper bound for the (weighted) number of singularities arises from applying Bézout's Theorem to the intersection of two generic polars of $C$:

$$\sum_{z\in C} \mu(C,z)\leq (d-1)^2,$$

$\mu(C,z) =\dim_\mathbf{C}\mathbf{C}\{x,y\}/(f_x,f_y)$ the Milnor number of $C$ at $z$.

On the other hand, in the case of arbitrary topological types $S_i$, we have the following existence theorem, which is asymptotically optimal (with respect to the occurring invariants and the exponent of $d$)

Theorem: [GLS,Lo]. $V_d(S_1, \dots, S_r) \not=\emptyset$ if $\sum^r_{i=1} \mu(S_i) \le \frac{1}{46}(d+2)^2$ and two additional conditions for the five ``worst'' singularities hold true.

In case of only one singularity we have the slightly better sufficient condition for existence, $\mu(S_1) \le \frac{1}{29}(d-5)^2$.

The theorem is just an existence statement, the proof gives no hint how to produce any equation. To produce explicit equations one needs some constructive method. Then the computer can be used in order to check the construction, or even, to improve the results. The following is a prominent example (actually, it belongs to a series of ``world record'' examples):

Example: [GN] The irreducible curve with affine equation

$$y^2-\,2y\bigl(x^{28}+2x^{21}y^{16}-2x^{14}y^{32}+4x^7y^{48}- 10y^{64}\bigr) +x^{56}+4x^{49}y^{16}=\,0$$

has degree 65 and an $A_{2260}$-singularity $(x^2-y^{2261} = 0)$ and a semiquasihomogeneous singularity $S_{9,16}$ with principal part $f_0=x^9+y^{16}$ as only singularities. In particular, it is an element of the variety $V_{65}(A_{2260},S_{9,16})$ which has negative expected dimension (hence is not T-smooth).

In order to verify this, one may proceed, using SINGULAR, as follows:

> ring s = 0,(x,y),ds;
> poly f = y2-2x28y-4x21y17+4x14y33-8x7y49+20y65+x56+4x49y16;
> matrix Hess = jacob(jacob(f));     //the Hessian matrix of f
> vdim(std(jacob(f)));               //the Milnor number of f
2260

Since the rank of the Hessian at 0 is checked to be 1, $f$ has an $A_k$ singularity at 0; it is an $A_{2260}$-singularity since the Milnor number is 2260. In the following we show that the projective curve defined by $f$ has no further singularities in the affine part. This follows from

$$\dim_\mathbf{C}(\mathbf{C}[x,y]_{\langle x,y\rangle}/\langl... ...m_\mathbf{C}(\mathbf{C}[x,y]/\langle \mbox{jacob}(f),f\rangle,$$

confirmed by SINGULAR:

> vdim(std(jacob(f)+f));
2260           // multiplicity of Sing(C) at 0 (local ordering)
> ring r = 0,(x,y),dp;
> poly f = fetch(s,f);
> vdim(std(jacob(f)+f));
2260           // multiplicity of Sing(C)      (global ordering)

Finally, we have to consider the singularities at infinity:

> ring sh = 0,(x,y,z),dp;
> poly f = fetch(s,f);
> poly F = homog(f,z); F;   // homogeneous polynomial defining C
4x49y16+20y65+x56z9-8x7y49z9+4x14y33z18-4x21y17z27-2x28yz36+y2z63
> ring r1 = 0,(y,z),dp;
> map phi = sh,1,y,z;
> poly g = phi(F);       // F in affine chart (x=1)
> vdim(std(jacob(g)+g));
120
> ring r2 = 0,(y,z),ds;  // local ring at (1:0:0)
> poly g = fetch(r1,g); g;
z9+4y16-2yz36-4y17z27+4y33z18-8y49z9+20y65+y2z63
> vdim(std(jacob(g)+g));
120

As before, we can conclude that there is precisely one singularity of $C$ on the line at infinity, situated at $(1:0:0)$, being semiquasihomogeneous of type $S_{9,16}$. (Note that in our computation we have considered all points at infinity except $(0:1:0)$. The latter is obviously not a point of $C$).

In the following we should like to mention a few problems and conjectures which are currently in the centre of research in connection with singular curves in $\P ^2_{\mathbf{C}}$.

Computing zero-dimensional ideals

Many of the questions concerning plane projective curves with prescribed singularities can be translated to properties of zero-dimensional (homogeneous) ideals $I\subset \mathbf{C}[x,y,z]$, e.g.,

• existence of curves with (ordinary) multiple points in prescribed position, or, more generally, existence of curves with prescribed position of infinitely near points (clusters),
• T-smoothness of the varieties $V_d(S_1, \dots, S_r),$
• existence of (global) deformations of projective curves.

For instance, consider the following problem: given points $p_1,\dots,p_n\in \P ^2_\mathbf{C}$ and positive integers $m_1,\dots,m_n$. Determine the dimension of the variety of curves of any degree $d$ passing through each of the points $p_i$ with multiplicity (at least) $m_i$, $i = 1, \dots, n$. The equivalent formulation would be: determine the ideal

$$I=\mathfrak{m}_{p_1}^{m_1} \cap \ldots \cap \mathfrak{m}_{p_n}^{m_n}\subset \mathbf{C}[x,y,z]\,,$$

$\mathfrak{m}_{p_i}$ the maximal ideal at $p_i$, and compute the Hilbert function $H_I$ of I.

Harbourne-Hirschowitz conjecture: Let $n>9$, $p_1,\dots,p_n\in \P ^2_\mathbf{C}$ in general position, $m$ a positive integer, and let $I=\mathfrak{m}_{p_1}^{m} \cap \ldots \cap \mathfrak{m}_{p_n}^{m}$. Then the Hilbert function satisfies

$$H_I(d)\,=\, \max \left\{0,\, \frac{(d\:+\:1)(d\:+\:2)}{2}- n \cdot\frac{m(m\:+\:1)}{2}\right\}$$

for all $d>0$. In other words, the variety of curves with n singular points of multiplicity (at least) $m$ at the prescribed (generic) points has the expected dimension.

There are several special cases where this conjecture is known to hold true; in particular, C. Ciliberto and R. Miranda [CM] have proven that it always holds for $m\leq 12$. Nevertheless the general conjecture is still far from being proven.

Nagata conjecture: Let $n>9$, $p_1,\dots,p_n\in \P ^2_\mathbf{C}$ in general position, $m_1,\dots,m_n$ positive integers, and let $a(m_1,\dots,m_n)$ denote the minimal degree of a curve passing through each of the points $p_i$ with multiplicity (at least) $m_i$, $i = 1, \dots, n$. Then

$$a(m_1,\dots,m_n) > \frac{m_1+\ldots + m_n}{\sqrt{n}}\,.$$

N. Nagata [Na] has proven the statement to be true for any $n>9$ being a square. There are many people working to prove this conjecture for other integers $n$ ([Ra]), or, at least, to improve the known lower bounds for $a(m_1,\dots,m_n)$ (the best known general bound is probably given in [Ro]). But the general question is still widely open.

Computer algebra could be used to provide evidence for such conjectures (or, to produce counter examples) provided one can solve the following problems:

1. find algorithms to compute the 0-dimensional ideals (related to the above problems). In many cases this is easy but for others this is unknown (e.g., to compute the equisingularity ideal for a sufficiently general singularity);

2. find fast algorithms to compute the intersection of zero-dimensional ideals. The general method for computing intersections via syzygies or elimination is too slow, due to the high complexity of the algorithms involved. There is already some considerable progress made, by the so-called Buchberger-Möller algorithm and further generalisations (cf. [AKR]), but certainly this is not yet sufficient.